{"id":65,"date":"2019-08-20T17:01:54","date_gmt":"2019-08-20T21:01:54","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/inverse-functions\/"},"modified":"2022-06-01T10:39:24","modified_gmt":"2022-06-01T14:39:24","slug":"inverse-functions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/inverse-functions\/","title":{"raw":"Inverse Functions","rendered":"Inverse Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Verify inverse functions.<\/li>\n \t<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.<\/li>\n \t<li>Find or evaluate the inverse of a function.<\/li>\n \t<li>Use the graph of a one-to-one function to graph its inverse function on the same axes.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135358875\">A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.<\/p>\n<p id=\"fs-id1165135701544\">If some physical machines can run in two directions, we might ask whether some of the function \u201cmachines\u201d we have been studying can also run backwards. <a class=\"autogenerated-content\" href=\"#Figure_01_07_001\">(Figure)<\/a> provides a visual representation of this question. In this section, we will consider the reverse nature of functions.<\/p>\n\n<div id=\"Figure_01_07_001\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141917\/CNX_Precalc_Figure_01_07_001.jpg\" alt=\"Diagram of a function and would be its inverse.\" width=\"731\" height=\"305\"> <strong>Figure 1. <\/strong>Can a function \u201cmachine\u201d operate in reverse?[\/caption]\n\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<div id=\"fs-id1165137725994\" class=\"bc-section section\">\n<h3>Verifying That Two Functions Are Inverse Functions<\/h3>\n<p id=\"fs-id1165135705795\">Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the <span class=\"no-emphasis\">Celsius<\/span> scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees <span class=\"no-emphasis\">Fahrenheit<\/span> to degrees Celsius. She finds the formula<\/p>\n\n<div id=\"fs-id1165137807176\" class=\"unnumbered aligncenter\">[latex]C=\\frac{5}{9}\\left(F-32\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135433486\">and substitutes 75 for[latex]\\,F\\,[\/latex]to calculate<\/p>\n\n<div id=\"fs-id1165137911210\" class=\"unnumbered aligncenter\">[latex]\\frac{5}{9}\\left(75-32\\right)\\approx 24\\text{\u00b0C}[\/latex]<\/div>\n<p id=\"fs-id1165137409312\">Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week\u2019s weather forecast from <a class=\"autogenerated-content\" href=\"#Figure_01_07_002\">(Figure)<\/a> for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.<\/p>\n\n<div id=\"Figure_01_07_002\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141923\/CNX_Precalc_Figure_01_07_002.jpg\" alt=\"A forecast of Monday\u2019s through Thursday\u2019s weather.\" width=\"731\" height=\"226\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\nAt first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for[latex]\\,F\\,[\/latex]after substituting a value for[latex]\\,C.\\,[\/latex]For example, to convert 26 degrees Celsius, she could write\n<div id=\"fs-id1165135548255\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 26&amp; =&amp; \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\ \\hfill 26\\cdot \\frac{9}{5}&amp; =&amp; F-32\\hfill \\\\ \\hfill F&amp; =&amp; 26\\cdot \\frac{9}{5}+32\\approx 79\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137540705\">After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.<\/p>\n<p id=\"fs-id1165137827441\">The formula for which Betty is searching corresponds to the idea of an <strong>inverse function<\/strong>, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.<\/p>\n<p id=\"fs-id1165135528385\">Given a function[latex]\\,f\\left(x\\right),\\,[\/latex]we represent its inverse as[latex]\\,{f}^{-1}\\left(x\\right),\\,[\/latex]read as[latex]\\,\u201cf\\,[\/latex]inverse of[latex]\\,x.\\text{\u201d}\\,[\/latex]The raised[latex]\\,-1\\,[\/latex]is part of the notation. It is not an exponent; it does not imply a power of[latex]\\,-1\\,[\/latex]. In other words,[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]does <em>not<\/em> mean[latex]\\,\\frac{1}{f\\left(x\\right)}\\,[\/latex]because[latex]\\,\\frac{1}{f\\left(x\\right)}\\,[\/latex]is the reciprocal of[latex]\\,f\\,[\/latex]and not the inverse.<\/p>\n<p id=\"fs-id1165137724926\">The \u201cexponent-like\u201d notation comes from an analogy between function composition and multiplication: just as[latex]\\,{a}^{-1}a=1\\,[\/latex](1 is the identity element for multiplication) for any nonzero number[latex]\\,a,\\,[\/latex]so[latex]\\,{f}^{-1}\\circ f\\,[\/latex]equals the identity function, that is,<\/p>\n\n<div id=\"fs-id1165134302408\" class=\"unnumbered aligncenter\">[latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(f\\left(x\\right)\\right)={f}^{-1}\\left(y\\right)=x[\/latex]<\/div>\n<p id=\"fs-id1165135667832\">This holds for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f.\\,[\/latex]Informally, this means that inverse functions \u201cundo\u201d each other. However, just as zero does not have a <span class=\"no-emphasis\">reciprocal<\/span>, some functions do not have inverses.<\/p>\n<p id=\"fs-id1165137655153\">Given a function[latex]\\,f\\left(x\\right),\\,[\/latex]we can verify whether some other function[latex]\\,g\\left(x\\right)\\,[\/latex]is the inverse of[latex]\\,f\\left(x\\right)\\,[\/latex]by checking whether either[latex]\\,g\\left(f\\left(x\\right)\\right)=x\\,[\/latex]or[latex]\\,f\\left(g\\left(x\\right)\\right)=x\\,[\/latex]is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)<\/p>\n<p id=\"fs-id1165135397975\">For example,[latex]\\,y=4x\\,[\/latex]and[latex]\\,y=\\frac{1}{4}x\\,[\/latex]are inverse functions.<\/p>\n\n<div id=\"fs-id1165137756798\" class=\"unnumbered aligncenter\">[latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(4x\\right)=\\frac{1}{4}\\left(4x\\right)=x[\/latex]<\/div>\n<p id=\"fs-id1165137767233\">and<\/p>\n\n<div id=\"fs-id1165137755853\" class=\"unnumbered aligncenter\">[latex]\\left({f}^{}\\circ {f}^{-1}\\right)\\left(x\\right)=f\\left(\\frac{1}{4}x\\right)=4\\left(\\frac{1}{4}x\\right)=x[\/latex]<\/div>\n<p id=\"fs-id1165137438777\">A few coordinate pairs from the graph of the function[latex]\\,y=4x\\,[\/latex]are (\u22122, \u22128), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function[latex]\\,y=\\frac{1}{4}x\\,[\/latex]are (\u22128, \u22122), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.<\/p>\n\n<div id=\"fs-id1165137933105\" class=\"textbox key-takeaways\">\n<h3>Inverse Function<\/h3>\n<p id=\"fs-id1165137473076\">For any <span class=\"no-emphasis\">one-to-one function<\/span>[latex]\\,f\\left(x\\right)=y,\\,[\/latex]a function[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]is an inverse function of[latex]\\,f\\,[\/latex]if[latex]\\,{f}^{-1}\\left(y\\right)=x.\\,[\/latex]This can also be written as[latex]\\,{f}^{-1}\\left(f\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f.\\,[\/latex]It also follows that[latex]\\,f\\left({f}^{-1}\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,{f}^{-1}\\,[\/latex]if[latex]\\,{f}^{-1}\\,[\/latex]is the inverse of[latex]\\,f.\\,[\/latex]<\/p>\n<p id=\"fs-id1165137444821\">The notation [latex]{f}^{-1}[\/latex] is read [latex]\\text{\u201c}f[\/latex] inverse.\u201d Like any other function, we can use any variable name as the input for [latex]{f}^{-1},[\/latex] so we will often write[latex]\\,{f}^{-1}\\left(x\\right),[\/latex] which we read as [latex]\u201cf[\/latex] inverse of [latex]x.\u201d[\/latex]\nKeep in mind that<\/p>\n\n<div id=\"fs-id1165137581324\" class=\"unnumbered aligncenter\">[latex]{f}^{-1}\\left(x\\right)\\ne \\frac{1}{f\\left(x\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165135194095\">and not all functions have inverses.<\/p>\n\n<\/div>\n<div id=\"Example_01_07_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137656641\">\n<div id=\"fs-id1165137922642\">\n<h3>Identifying an Inverse Function for a Given Input-Output Pair<\/h3>\n<p id=\"fs-id1165137659325\">If for a particular one-to-one function[latex]\\,f\\left(2\\right)=4\\,[\/latex]and[latex]\\,f\\left(5\\right)=12,\\,[\/latex]what are the corresponding input and output values for the inverse function?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137768306\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137768306\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137768306\"]\n<p id=\"fs-id1165137737081\">The inverse function reverses the input and output quantities, so if<\/p>\n\n<div id=\"fs-id1165137462459\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill f\\left(2\\right)&amp; =&amp; 4,\\text{ then }{f}^{-1}\\left(4\\right)=2;\\hfill \\\\ \\hfill f\\left(5\\right)&amp; =&amp; 12,{\\text{ then f}}^{-1}\\left(12\\right)=5.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137659464\">Alternatively, if we want to name the inverse function[latex]\\,g,\\,[\/latex]then[latex]\\,g\\left(4\\right)=2\\,[\/latex]and[latex]\\,g\\left(12\\right)=5.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135245520\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135508518\">Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See <a class=\"autogenerated-content\" href=\"#Table_01_07_01\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_01_07_01\" summary=\"For (x,f(x)) we have the values (2, 4) and (5, 12); for (x, g(x)), we have the values (4, 2) and (12, 5).\">\n<thead>\n<tr>\n<th>[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\n<th>[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,12\\right)[\/latex]<\/td>\n<td>[latex]\\left(12,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635377\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_01\">\n<div id=\"fs-id1165137659088\">\n<p id=\"fs-id1165137659089\">Given that[latex]\\,{h}^{-1}\\left(6\\right)=2,\\,[\/latex]what are the corresponding input and output values of the original function[latex]\\,h?\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137645907\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137645907\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137645907\"]\n<p id=\"fs-id1165137645908\">[latex]h\\left(2\\right)=6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134357354\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135434077\"><strong>Given two functions[latex]\\,\\,f\\left(x\\right)\\,\\,[\/latex]and[latex]\\,g\\left(x\\right),\\,[\/latex]test whether the functions are inverses of each other.<\/strong><\/p>\n\n<ol id=\"fs-id1165137452358\" type=\"1\">\n \t<li>Determine whether[latex]\\,f\\left(g\\left(x\\right)\\right)=x\\,[\/latex]or[latex]\\,g\\left(f\\left(x\\right)\\right)=x.[\/latex]<\/li>\n \t<li>If either statement is true, then both are true, and[latex]\\,g={f}^{-1}\\,[\/latex]and[latex]\\,f={g}^{-1}.\\,[\/latex]If either statement is false, then both are false, and[latex]\\,g\\ne {f}^{-1}\\,[\/latex]and[latex]\\,f\\ne {g}^{-1}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137557051\">\n<div id=\"fs-id1165137679032\">\n<h3>Testing Inverse Relationships Algebraically<\/h3>\n<p id=\"fs-id1165135519417\">If[latex]\\,f\\left(x\\right)=\\frac{1}{x+2}\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\frac{1}{x}-2,\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137627632\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137627632\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137627632\"]\n<div id=\"fs-id1165137675509\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill g\\left(f\\left(x\\right)\\right)&amp; =&amp; \\frac{1}{\\left(\\frac{1}{x+2}\\right)}-2\\hfill \\\\ &amp; =&amp; x+2-2\\hfill \\\\ &amp; =&amp; x\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137611481\">so<\/p>\n\n<div id=\"fs-id1165135678636\" class=\"unnumbered aligncenter\">[latex]g={f}^{-1}\\text{ and }f={g}^{-1}[\/latex]<\/div>\n<p id=\"fs-id1165135436648\">This is enough to answer yes to the question, but we can also verify the other formula.<\/p>\n\n<div id=\"fs-id1165137784350\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill f\\left(g\\left(x\\right)\\right)&amp; =&amp; \\frac{1}{\\frac{1}{x}-2+2}\\hfill \\\\ &amp; =&amp; \\frac{1}{\\frac{1}{x}}\\hfill \\\\ &amp; =&amp; x\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137733685\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135389000\">Notice the inverse operations are in reverse order of the operations from the original function.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911663\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_02\">\n<div id=\"fs-id1165135160549\">\n<p id=\"fs-id1165135160550\">If[latex]\\,f\\left(x\\right)={x}^{3}-4\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\sqrt[\\,3]{x+4},\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137456449\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137456449\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137456449\"]\n<p id=\"fs-id1165137600434\">Yes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135259560\">\n<div id=\"fs-id1165134042918\">\n<h3>Determining Inverse Relationships for Power Functions<\/h3>\n<p id=\"fs-id1165137441834\">If[latex]\\,f\\left(x\\right)={x}^{3}\\,[\/latex](the cube function) and[latex]\\,g\\left(x\\right)=\\frac{1}{3}x,\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137442603\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137442603\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137442603\"]\n<div id=\"fs-id1165137591632\" class=\"unnumbered aligncenter\">[latex]f\\left(g\\left(x\\right)\\right)=\\frac{{x}^{3}}{27}\\ne x[\/latex]<\/div>\n<p id=\"fs-id1165137694053\">No, the functions are not inverses.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165135317479\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134192978\">The correct inverse to the cube is, of course, the cube root[latex]\\,\\sqrt[3]{x}={x}^{\\frac{1}{3}},\\,[\/latex]that is, the one-third is an exponent, not a multiplier.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806084\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_03\">\n<div id=\"fs-id1165135195489\">\n<p id=\"fs-id1165137573532\">If[latex]\\,f\\left(x\\right)={\\left(x-1\\right)}^{3}\\,\\text{and}\\,g\\left(x\\right)=\\sqrt[3]{x}+1,\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137501356\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137501356\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137501356\"]\n<p id=\"fs-id1165137662080\">Yes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660004\" class=\"bc-section section\">\n<h3>Finding Domain and Range of Inverse Functions<\/h3>\n<p id=\"fs-id1165137591020\">The outputs of the function[latex]\\,f\\,[\/latex]are the inputs to[latex]\\,{f}^{-1},\\,[\/latex]so the range of[latex]\\,f\\,[\/latex]is also the domain of[latex]\\,{f}^{-1}.\\,[\/latex]Likewise, because the inputs to[latex]\\,f\\,[\/latex]are the outputs of[latex]\\,{f}^{-1},\\,[\/latex]the domain of[latex]\\,f\\,[\/latex]is the range of[latex]\\,{f}^{-1}.\\,[\/latex]We can visualize the situation as in <a class=\"autogenerated-content\" href=\"#Figure_01_07_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141943\/CNX_Precalc_Figure_01_07_003.jpg\" alt=\"Domain and range of a function and its inverse.\" width=\"487\" height=\"143\"> <strong>Figure 3. <\/strong>Domain and range of a function and its inverse[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135557891\">When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of[latex]\\,f\\left(x\\right)=\\sqrt{x}\\,[\/latex]is[latex]\\,{f}^{-1}\\left(x\\right)={x}^{2},\\,[\/latex]because a square \u201cundoes\u201d a square root; but the square is only the inverse of the square root on the domain[latex]\\,\\left[0,\\infty \\right),\\,[\/latex]since that is the range of[latex]\\,f\\left(x\\right)=\\sqrt{x}.[\/latex]<\/p>\n<p id=\"fs-id1165137730185\">We can look at this problem from the other side, starting with the square (toolkit quadratic) function[latex]\\,f\\left(x\\right)={x}^{2}.\\,[\/latex]If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and \u20133. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the \u201cinverse\u201d is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.<\/p>\n<p id=\"fs-id1165137823552\">In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]with its domain limited to[latex]\\,\\left[0,\\infty \\right),\\,[\/latex]which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).<\/p>\n<p id=\"fs-id1165132037000\">If[latex]\\,f\\left(x\\right)={\\left(x-1\\right)}^{2}\\,[\/latex]on[latex]\\,\\left[1,\\infty \\right),\\,[\/latex]then the inverse function is[latex]\\,{f}^{-1}\\left(x\\right)=\\sqrt{x}+1.[\/latex]<\/p>\n\n<ul id=\"fs-id1165137851227\">\n \t<li>The domain of[latex]\\,f\\,[\/latex]= range of[latex]\\,{f}^{-1}\\,[\/latex]=[latex]\\,\\left[1,\\infty \\right).[\/latex]<\/li>\n \t<li>The domain of[latex]\\,{f}^{-1}\\,[\/latex]= range of[latex]\\,f\\,[\/latex]=[latex]\\,\\left[0,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<div id=\"fs-id1165137733804\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137723526\"><strong>Is it possible for a function to have more than one inverse?<\/strong><\/p>\n<p id=\"fs-id1165137456608\"><em>No. If two supposedly different functions, say,[latex]\\,g\\,[\/latex]and[latex]\\,h,\\,[\/latex]both meet the definition of being inverses of another function[latex]\\,f,\\,[\/latex]then you can prove that[latex]\\,g=h.\\,[\/latex]We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165137704938\" class=\"textbox key-takeaways\">\n<h3>Domain and Range of Inverse Functions<\/h3>\n<p id=\"fs-id1165135319550\">The range of a function[latex]\\,f\\left(x\\right)\\,[\/latex]is the domain of the inverse function[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137673886\">The domain of[latex]\\,f\\left(x\\right)\\,[\/latex]is the range of[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135308785\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137605040\"><strong>Given a function, find the domain and range of its inverse.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165137530434\" type=\"1\">\n \t<li>If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse.<\/li>\n \t<li>If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_05\" class=\"textbox examples\">\n<h3>Finding the Inverses of Toolkit Functions<\/h3>\n<p id=\"fs-id1165137448020\">Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in <a class=\"autogenerated-content\" href=\"#Table_01_07_02\">(Figure)<\/a>. We restrict the domain in such a fashion that the function assumes all <em>y<\/em>-values exactly once.<\/p>\n\n<table id=\"Table_01_07_02\" summary=\"A list of the toolkit function. The constant function is f(x) = c where c is the constant; the identity function is f(x) = x; the absolute function is f(x)=|x|; the quadratic function is f(x) = x^2; the cubic function is f(x)=x^3; the reciprocal function is f(x)=1\/x; the reciprocal squared function is f(x)=1\/x^2; the square root function is f(x)=sqrt(x); the cube root function is f(x) = x^(1\/3).\"><colgroup> <col> <col> <col> <col> <col><\/colgroup>\n<thead>\n<tr>\n<th>Constant<\/th>\n<th>Identity<\/th>\n<th>Quadratic<\/th>\n<th>Cubic<\/th>\n<th>Reciprocal<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f\\left(x\\right)=c[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Reciprocal squared<\/strong><\/td>\n<td><strong>Cube root<\/strong><\/td>\n<td><strong>Square root<\/strong><\/td>\n<td><strong>Absolute value<\/strong><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137767030\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137767030\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137767030\"]\n<p id=\"fs-id1165132988445\">The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no inverse.<\/p>\nThe absolute value function can be restricted to the domain[latex]\\,\\left[0,\\infty \\right),[\/latex]where it is equal to the identity function.\n<p id=\"fs-id1165137642849\">The reciprocal-squared function can be restricted to the domain[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165137901280\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137742302\">We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_004\">(Figure)<\/a>. They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes the horizontal line test, then in that restricted domain, it can have an inverse.<\/p>\n\n<div id=\"Figure_01_07_004\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141957\/CNX_Precalc_Figure_01_07_004ab.jpg\" alt=\"Graph of an absolute function.\" width=\"975\" height=\"404\"> <strong>Figure 4.<\/strong> (a) Absolute value (b) Reciprocal square[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137544599\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_09\">\n<div id=\"fs-id1165135406939\">\n<p id=\"fs-id1165137507853\">The domain of function[latex]\\,f\\,[\/latex]is[latex]\\,\\left(1,\\infty \\right)\\,[\/latex]and the range of function[latex]\\,f\\,[\/latex]is[latex]\\,\\left(\\mathrm{-\\infty },-2\\right).\\,[\/latex]Find the domain and range of the inverse function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137431277\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137431277\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137431277\"]\n<p id=\"fs-id1165137451713\">The domain of function[latex]\\,{f}^{-1}\\,[\/latex]is[latex]\\,\\left(-\\infty \\text{,}-2\\right)\\,[\/latex]and the range of function[latex]\\,{f}^{-1}\\,[\/latex]is[latex]\\,\\left(1,\\infty \\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619159\" class=\"bc-section section\">\n<h3>Finding and Evaluating Inverse Functions<\/h3>\n<p id=\"fs-id1165137761017\">Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.<\/p>\n\n<div id=\"fs-id1165135466392\" class=\"bc-section section\">\n<h4>Inverting Tabular Functions<\/h4>\n<p id=\"fs-id1165135190714\">Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.<\/p>\n<p id=\"fs-id1165137422578\">Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.<\/p>\n\n<div id=\"Example_01_07_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135544995\">\n<div id=\"fs-id1165137698262\">\n<h3>Interpreting the Inverse of a Tabular Function<\/h3>\n<p id=\"fs-id1165135435474\">A function[latex]\\,f\\left(t\\right)\\,[\/latex]is given in <a class=\"autogenerated-content\" href=\"#Table_01_07_03\">(Figure)<\/a>, showing distance in miles that a car has traveled in[latex]\\,t\\,[\/latex]minutes. Find and interpret[latex]\\,{f}^{-1}\\left(70\\right).[\/latex]<\/p>\n\n<table id=\"Table_01_07_03\" summary=\"Two rows and five columns. The first row is labeled \u201ct (minutes)\u201d, and the second row is labeled \u201cf(x) (miles)\u201d. Reading the columns as ordered pairs, we have the following values (30, 20), (50, 40), (70, 60), and (90, 70).\"><colgroup> <col> <col> <col> <col> <col><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]t\\text{ (minutes)}[\/latex]<\/strong><\/td>\n<td>30<\/td>\n<td>50<\/td>\n<td>70<\/td>\n<td>90<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex]<\/strong><\/td>\n<td>20<\/td>\n<td>40<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137407569\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137407569\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137407569\"]\n<p id=\"fs-id1165137640334\">The inverse function takes an output of[latex]\\,f\\,[\/latex]and returns an input for[latex]\\,f.\\,[\/latex]So in the expression[latex]\\,{f}^{-1}\\left(70\\right),\\,[\/latex]70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function[latex]\\,f,\\,[\/latex]90 minutes, so[latex]\\,{f}^{-1}\\left(70\\right)=90.\\,[\/latex]The interpretation of this is that, to drive 70 miles, it took 90 minutes.<\/p>\n<p id=\"fs-id1165135181841\">Alternatively, recall that the definition of the inverse was that if[latex]\\,f\\left(a\\right)=b,\\,[\/latex]then[latex]\\,{f}^{-1}\\left(b\\right)=a.\\,[\/latex]By this definition, if we are given[latex]\\,{f}^{-1}\\left(70\\right)=a,\\,[\/latex]then we are looking for a value[latex]\\,a\\,[\/latex]so that[latex]\\,f\\left(a\\right)=70.\\,[\/latex]In this case, we are looking for a[latex]\\,t\\,[\/latex]so that[latex]\\,f\\left(t\\right)=70,\\,[\/latex]which is when[latex]\\,t=90.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135169494\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_04\">\n<div id=\"fs-id1165135443767\">\n<p id=\"fs-id1165134108483\">Using <a class=\"autogenerated-content\" href=\"#Table_01_07_04\">(Figure)<\/a>, find and interpret (a)[latex]\\text{ }f\\left(60\\right),[\/latex]and (b)[latex]\\text{ }{f}^{-1}\\left(60\\right).[\/latex]<\/p>\n\n<table id=\"Table_01_07_04\" summary=\"Two rows and five columns. The first row is labeled \u201ct (minutes)\u201d, and the second row is labeled \u201cf(t)\u201d. Reading the columns as ordered pairs, we have the following values (30, 20), (50, 40), (70, 60), and (90, 70).\"><colgroup> <col> <col> <col> <col> <col> <col><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]t\\text{ (minutes)}[\/latex]<\/strong><\/td>\n<td>30<\/td>\n<td>50<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<td>90<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex]<\/strong><\/td>\n<td>20<\/td>\n<td>40<\/td>\n<td>50<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137862841\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137862841\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137862841\"]\n<ol id=\"fs-id1165137862843\" type=\"a\">\n \t<li>[latex]f\\left(60\\right)=50.\\,[\/latex]In 60 minutes, 50 miles are traveled.<\/li>\n \t<li>[latex]{f}^{-1}\\left(60\\right)=70.\\,[\/latex]To travel 60 miles, it will take 70 minutes.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137418615\" class=\"bc-section section\">\n<h4>Evaluating the Inverse of a Function, Given a Graph of the Original Function<\/h4>\n<p id=\"fs-id1165137400045\">We saw in <a class=\"target-chapter\" href=\"\/contents\/55f2e8ec-a982-4586-9d48-a2f43d7b4107\">Functions and Function Notation<\/a> that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the <em>vertical<\/em> extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the <em>horizontal<\/em> extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function\u2019s graph.<\/p>\n\n<div id=\"fs-id1165133045388\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135333128\"><strong>Given the graph of a function, evaluate its inverse at specific points.<\/strong><\/p>\n\n<ol id=\"fs-id1165137464840\" type=\"1\">\n \t<li>Find the desired input on the <em>y<\/em>-axis of the given graph.<\/li>\n \t<li>Read the inverse function\u2019s output from the <em>x<\/em>-axis of the given graph.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_07\" class=\"textbox examples\">\n<div id=\"fs-id1165135434803\">\n<div id=\"fs-id1165135434805\">\n<h3>Evaluating a Function and Its Inverse from a Graph at Specific Points<\/h3>\n<p id=\"fs-id1165134108622\">A function[latex]\\,g\\left(x\\right)\\,[\/latex]is given in <a class=\"autogenerated-content\" href=\"#Figure_01_07_006\">(Figure)<\/a>. Find[latex]\\,g\\left(3\\right)\\,[\/latex]and[latex]\\,{g}^{-1}\\left(3\\right).[\/latex]<\/p>\n\n<div id=\"Figure_01_07_006\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142008\/CNX_Precalc_Figure_01_07_006.jpg\" alt=\"Graph of g(x).\" width=\"487\" height=\"254\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137468840\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137468840\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137468840\"]\n<p id=\"fs-id1165137468842\">To evaluate [latex]g\\left(3\\right),\\,[\/latex]we find 3 on the <em>x<\/em>-axis and find the corresponding output value on the <em>y<\/em>-axis. The point [latex]\\,\\left(3,1\\right)\\,[\/latex]tells us that[latex]\\,g\\left(3\\right)=1.[\/latex]<\/p>\n<p id=\"fs-id1165137405078\">To evaluate[latex]\\,{g}^{-1}\\left(3\\right),\\,[\/latex]recall that by definition[latex]\\,{g}^{-1}\\left(3\\right)\\,[\/latex]means the value of <em>x<\/em> for which[latex]\\,g\\left(x\\right)=3.\\,[\/latex]By looking for the output value 3 on the vertical axis, we find the point[latex]\\,\\left(5,3\\right)\\,[\/latex]on the graph, which means[latex]\\,g\\left(5\\right)=3,\\,[\/latex]so by definition,[latex]\\,{g}^{-1}\\left(3\\right)=5.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_01_07_007\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_007\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142010\/CNX_Precalc_Figure_01_07_007.jpg\" alt=\"Graph of g(x).\" width=\"487\" height=\"254\"> <strong>Figure 6.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137667918\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_05\">\n<div id=\"fs-id1165137812559\">\n<p id=\"fs-id1165137812560\">Using the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_07_007\">(Figure)<\/a>, (a) find[latex]\\,{g}^{-1}\\left(1\\right),[\/latex]and (b) estimate[latex]\\,{g}^{-1}\\left(4\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135528890\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135528890\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135528890\"]\n<p id=\"fs-id1165135528891\">a. 3; b. 5.6<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605437\" class=\"bc-section section\">\n<h4>Finding Inverses of Functions Represented by Formulas<\/h4>\n<p id=\"fs-id1165137433184\">Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula\u2014for example,[latex]\\,y\\,[\/latex]as a function of[latex]\\,x\\text{\u2014}[\/latex]we can often find the inverse function by solving to obtain[latex]\\,x\\,[\/latex]as a function of[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"fs-id1165137652548\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135195849\"><strong>Given a function represented by a formula, find the inverse.<\/strong><\/p>\n\n<ol id=\"fs-id1165135443898\" type=\"1\">\n \t<li>Make sure[latex]\\,f\\,[\/latex]is a one-to-one function.<\/li>\n \t<li>Solve for[latex]\\,x.[\/latex]<\/li>\n \t<li>Interchange[latex]\\,x\\,[\/latex]and[latex]\\,y.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_08\" class=\"textbox examples\">\n<div id=\"fs-id1165135186316\">\n<div id=\"fs-id1165135186318\">\n<h3>Inverting the Fahrenheit-to-Celsius Function<\/h3>\n<p id=\"fs-id1165137596585\">Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.<\/p>\n\n<div id=\"fs-id1165133306998\" class=\"unnumbered aligncenter\">[latex]C=\\frac{5}{9}\\left(F-32\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135417800\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135417800\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135417800\"]\n<div id=\"fs-id1165135193737\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill C&amp; =&amp; \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\ \\hfill C\\cdot \\frac{9}{5}&amp; =&amp; F-32\\hfill \\\\ \\hfill F&amp; =&amp; \\frac{9}{5}C+32\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137819987\">By solving in general, we have uncovered the inverse function. If<\/p>\n\n<div id=\"fs-id1165135173380\" class=\"unnumbered aligncenter\">[latex]C=h\\left(F\\right)=\\frac{5}{9}\\left(F-32\\right),[\/latex]<\/div>\n<p id=\"fs-id1165135435603\">then<\/p>\n\n<div id=\"fs-id1165137772327\" class=\"unnumbered aligncenter\">[latex]F={h}^{-1}\\left(C\\right)=\\frac{9}{5}C+32[\/latex]<\/div>\n<p id=\"fs-id1165137573279\">In this case, we introduced a function[latex]\\,h\\,[\/latex]to represent the conversion because the input and output variables are descriptive, and writing[latex]\\,{C}^{-1}\\,[\/latex]could get confusing.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_06\">\n<div id=\"fs-id1165135563330\">\n<p id=\"fs-id1165135563331\">Solve for[latex]\\,x\\,[\/latex]in terms of[latex]\\,y\\,[\/latex]given[latex]\\,y=\\frac{1}{3}\\left(x-5\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134049417\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134049417\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134049417\"]\n<p id=\"fs-id1165134049418\">[latex]x=3y+5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134065146\">\n<div id=\"fs-id1165137409366\">\n<h3>Solving to Find an Inverse Function<\/h3>\n<p id=\"fs-id1165137891504\">Find the inverse of the function[latex]\\,f\\left(x\\right)=\\frac{2}{x-3}+4.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137482074\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137482074\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137482074\"]\n<div id=\"fs-id1165135189953\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill y&amp; =&amp; \\frac{2}{x-3}+4\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Set up an equation}.\\hfill \\\\ \\hfill y-4&amp; =&amp; \\frac{2}{x-3}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Subtract 4 from both sides}.\\hfill \\\\ \\hfill x-3&amp; =&amp; \\frac{2}{y-4}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply both sides by }x-3\\text{ and divide by }y-4.\\hfill \\\\ \\hfill x&amp; =&amp; \\frac{2}{y-4}+3\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Add 3 to both sides}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137678168\">So[latex]\\,{f}^{-1}\\left(y\\right)=\\frac{2}{y-4}+3\\,[\/latex]or[latex]\\,{f}^{-1}\\left(x\\right)=\\frac{2}{x-4}+3.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137864156\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135394231\">The domain and range of[latex]\\,f\\,[\/latex]exclude the values 3 and 4, respectively.[latex]\\,f\\,[\/latex] and [latex]\\,{f}^{-1}\\,[\/latex]are equal at two points but are not the same function, as we can see by creating <a class=\"autogenerated-content\" href=\"#Table_01_07_05\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_01_07_05\" summary=\"The values of f(x) are: f(1)=3, f(2)=2, and f(5)=5. So f^(-1)(y)=y.\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<td>[latex]{f}^{-1}\\left(y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>3<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<td>[latex]y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137603677\">\n<div id=\"fs-id1165137547656\">\n<h3>Solving to Find an Inverse with Radicals<\/h3>\nFind the inverse of the function[latex]\\,f\\left(x\\right)=2+\\sqrt{x-4}.[\/latex]\n\n<\/div>\n<div id=\"fs-id1165135193684\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135193684\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135193684\"]\n<div id=\"fs-id1165137828173\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill y&amp; =&amp; 2+\\sqrt{x-4}\\hfill \\\\ \\hfill {\\left(y-2\\right)}^{2}&amp; =&amp; x-4\\hfill \\\\ \\hfill x&amp; =&amp; {\\left(y-2\\right)}^{2}+4\\hfill \\end{array}[\/latex]<\/div>\nSo[latex]\\,{f}^{-1}\\left(x\\right)={\\left(x-2\\right)}^{2}+4.[\/latex]\n<p id=\"fs-id1165137900392\">The domain of[latex]\\,f\\,[\/latex]is[latex]\\,\\left[4,\\infty \\right).\\,[\/latex]Notice that the range of[latex]\\,f\\,[\/latex]is[latex]\\,\\left[2,\\infty \\right),\\,[\/latex]so this means that the domain of the inverse function[latex]\\,{f}^{-1}\\,[\/latex]is also[latex]\\,\\left[2,\\infty \\right).[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137667328\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135546050\">The formula we found for[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]looks like it would be valid for all real[latex]\\,x.\\,[\/latex]However,[latex]\\,{f}^{-1}\\,[\/latex]itself must have an inverse (namely,[latex]\\,f\\,[\/latex]) so we have to restrict the domain of[latex]\\,{f}^{-1}\\,[\/latex]to[latex]\\,\\left[2,\\infty \\right)\\,[\/latex]in order to make[latex]\\,{f}^{-1}\\,[\/latex]a one-to-one function. This domain of[latex]\\,{f}^{-1}\\,[\/latex]is exactly the range of[latex]\\,f.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640068\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_07\">\n<div id=\"fs-id1165137756073\">\n<p id=\"fs-id1165137756074\">What is the inverse of the function[latex]\\,f\\left(x\\right)=2-\\sqrt{x}?[\/latex]State the domains of both the function and the inverse function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137937549\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137937549\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137937549\"]\n<p id=\"fs-id1165137937550\">[latex]{f}^{-1}\\left(x\\right)={\\left(2-x\\right)}^{2};\\,\\,\\text{domain}\\,\\,\\text{of}\\,\\,f:\\,\\,\\left[0,\\infty \\right);\\,\\,\\text{domain}\\,\\,\\text{of}\\,\\,{f}^{-1}:\\,\\,\\left(-\\infty ,2\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473011\" class=\"bc-section section\">\n<h3>Finding Inverse Functions and Their Graphs<\/h3>\n<p id=\"fs-id1165137463843\">Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]restricted to the domain[latex]\\,\\left[0,\\infty \\right)\\text{,}[\/latex] on which this function is one-to-one, and graph it as in <a class=\"autogenerated-content\" href=\"#Figure_01_07_008\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_008\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142016\/CNX_Precalc_Figure_01_07_008.jpg\" alt=\"Graph of f(x).\" width=\"487\" height=\"254\"> <strong>Figure 7. <\/strong>Quadratic function with domain restricted to [0, \u221e).[\/caption]<\/div>\n<p id=\"fs-id1165137419977\"><span class=\"no-emphasis\">Restricting the domain<\/span> to[latex]\\,\\left[0,\\infty \\right)\\,[\/latex]makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.<\/p>\n<p id=\"fs-id1165137656093\">We already know that the inverse of the toolkit quadratic function is the square root function, that is, [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}.[\/latex] What happens if we graph both [latex]f\\text{ }[\/latex] and [latex]{f}^{-1}[\/latex] on the same set of axes, using the [latex]x\\text{-}[\/latex]axis for the input to both [latex]f\\text{ and }{f}^{-1}?[\/latex]<\/p>\n<p id=\"fs-id1165131968090\">We notice a distinct relationship: The graph of[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]is the graph of[latex]\\,f\\left(x\\right)\\,[\/latex]reflected about the diagonal line[latex]\\,y=x,\\,[\/latex]which we will call the identity line, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_009\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_009\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142034\/CNX_Precalc_Figure_01_07_009.jpg\" alt=\"Graph of f(x) and f^(-1)(x).\" width=\"487\" height=\"251\"> <strong>Figure 8. <\/strong>Square and square-root functions on the non-negative domain[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137393212\">This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.<\/p>\n\n<div id=\"Example_01_07_11\" class=\"textbox examples\">\n<div id=\"fs-id1165134430460\">\n<div id=\"fs-id1165134430463\">\n<h3>Finding the Inverse of a Function Using Reflection about the Identity Line<\/h3>\n<p id=\"fs-id1165134342627\">Given the graph of[latex]\\,f\\left(x\\right)\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Figure_01_07_010\">(Figure)<\/a>, sketch a graph of[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n\n<div id=\"Figure_01_07_010\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142046\/CNX_Precalc_Figure_01_07_010.jpg\" alt=\"Graph of f^(-1)(x).\" width=\"487\" height=\"363\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407658\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137407658\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137407658\"]\n<p id=\"fs-id1165137407660\">This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of[latex]\\,\\left(0,\\infty \\right)\\,[\/latex]and range of[latex]\\,\\left(-\\infty ,\\infty \\right),\\,[\/latex]so the inverse will have a domain of[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]and range of[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/p>\nIf we reflect this graph over the line[latex]\\,y=x,\\,[\/latex]the point[latex]\\,\\left(1,0\\right)\\,[\/latex]reflects to[latex]\\,\\left(0,1\\right)\\,[\/latex]and the point[latex]\\,\\left(4,2\\right)\\,[\/latex]reflects to[latex]\\,\\left(2,4\\right).\\,[\/latex]Sketching the inverse on the same axes as the original graph gives <a class=\"autogenerated-content\" href=\"#Figure_01_07_011\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142103\/CNX_Precalc_Figure_01_07_011.jpg\" alt=\"Graph of f(x) and f^(-1)(x).\" width=\"487\" height=\"363\"> <strong>Figure 10. <\/strong>The function and its inverse, showing reflection about the identity line[\/caption]\n<p id=\"fs-id1165137416305\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187125\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_08\">\n<div id=\"fs-id1165137619929\">\n<p id=\"fs-id1165137619930\">Draw graphs of the functions[latex]\\,f\\text{ }[\/latex]and[latex]\\text{ }{f}^{-1}[\/latex]from <a class=\"autogenerated-content\" href=\"#Example_01_07_09\">(Figure)<\/a>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137911739\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137911739\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137911739\"]<span id=\"fs-id1165137539140\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142110\/CNX_Precalc_Figure_01_07_012.jpg\" alt=\"Graph of f(x) and f^(-1)(x).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627081\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165134388228\"><strong>Is there any function that is equal to its own inverse?<\/strong><\/p>\n<p id=\"fs-id1165137602656\"><em>Yes. If[latex]\\,f={f}^{-1},\\,[\/latex]then[latex]\\,f\\left(f\\left(x\\right)\\right)=x,\\,[\/latex]and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because<\/em><\/p>\n\n<div id=\"fs-id1165135205827\" class=\"unnumbered aligncenter\">[latex]\\frac{1}{\\frac{1}{x}}=x[\/latex]<\/div>\n<p id=\"fs-id1165137897050\"><em>Any function[latex]\\,f\\left(x\\right)=c-x,\\,[\/latex]where[latex]\\,c\\,[\/latex]is a constant, is also equal to its own inverse.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165137410410\" class=\"precalculus media\">\n<p id=\"fs-id1165137410501\">Access these online resources for additional instruction and practice with inverse functions.<\/p>\n\n<ul id=\"fs-id1165137582034\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/inversefunction\">Inverse Functions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/onetoone\">One-to-one Functions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/inversfuncgraph\">Inverse Function Values Using Graph<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/restrictdomain\">Restricting the Domain and Finding the Inverse<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-10\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC01\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137591826\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135536334\">\n \t<li>If[latex]\\,g\\left(x\\right)\\,[\/latex]is the inverse of[latex]\\,f\\left(x\\right),\\,[\/latex]then[latex]\\,g\\left(f\\left(x\\right)\\right)=f\\left(g\\left(x\\right)\\right)=x.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_01_07_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_01_07_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_01_07_03\">(Figure)<\/a>.<\/li>\n \t<li>Only some of the toolkit functions have an inverse. See <a class=\"autogenerated-content\" href=\"#Example_01_07_05\">(Figure)<\/a>.<\/li>\n \t<li>For a function to have an inverse, it must be one-to-one (pass the horizontal line test).<\/li>\n \t<li>A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.<\/li>\n \t<li>For a tabular function, exchange the input and output rows to obtain the inverse. See <a class=\"autogenerated-content\" href=\"#Example_01_07_06\">(Figure)<\/a>.<\/li>\n \t<li>The inverse of a function can be determined at specific points on its graph. See <a class=\"autogenerated-content\" href=\"#Example_01_07_07\">(Figure)<\/a>.<\/li>\n \t<li>To find the inverse of a formula, solve the equation[latex]\\,y=f\\left(x\\right)\\,[\/latex]for[latex]\\,x\\,[\/latex]as a function of[latex]\\,y.\\,[\/latex]Then exchange the labels[latex]\\,x\\,[\/latex]and[latex]\\,\\,y.\\,\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_01_07_08\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_01_07_09\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_01_07_10\">(Figure)<\/a>.<\/li>\n \t<li>The graph of an inverse function is the reflection of the graph of the original function across the line[latex]\\,y=x.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_01_07_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137871042\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135187563\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137407341\">\n<div id=\"fs-id1165135193086\">\n<p id=\"fs-id1165135193088\">Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137517264\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137517264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137517264\"]\n<p id=\"fs-id1165134061896\">Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that[latex]\\,y[\/latex]-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no[latex]\\,y[\/latex]-values repeat and the function is one-to-one.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137408636\">\n<div id=\"fs-id1165137408638\">\n<p id=\"fs-id1165134113962\">Why do we restrict the domain of the function[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]to find the function\u2019s inverse?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137389621\">\n<div id=\"fs-id1165135400199\">\n<p id=\"fs-id1165135400201\">Can a function be its own inverse? Explain.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137553896\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137553896\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137553896\"]\n<p id=\"fs-id1165137654653\">Yes. For example,[latex]\\,f\\left(x\\right)=\\frac{1}{x}\\,[\/latex]is its own inverse.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188794\">\n<div id=\"fs-id1165137564806\">\n<p id=\"fs-id1165137564808\">Are one-to-one functions either always increasing or always decreasing? Why or why not?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419050\">\n<div id=\"fs-id1165137932403\">\n<p id=\"fs-id1165137932405\">How do you find the inverse of a function algebraically?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137785042\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137785042\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137785042\"]\n<p id=\"fs-id1165137673500\">Given a function[latex]\\,y=f\\left(x\\right),\\,[\/latex]solve for[latex]\\,x\\,[\/latex]in terms of[latex]\\,y.\\,[\/latex]Interchange the[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]Solve the new equation for[latex]\\,y.\\,[\/latex]The expression for[latex]\\,y\\,[\/latex]is the inverse,[latex]\\,y={f}^{-1}\\left(x\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836714\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165137422830\">\n<div id=\"fs-id1165137806758\">\n<p id=\"fs-id1165137806761\">Show that the function[latex]\\,f\\left(x\\right)=a-x\\,[\/latex]is its own inverse for all real numbers[latex]\\,a.\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137469451\">For the following exercises, find[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]for each function.<\/p>\n\n<div id=\"fs-id1165134312158\">\n<div id=\"fs-id1165134312161\">\n<p id=\"fs-id1165137562307\">[latex]f\\left(x\\right)=x+3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135196794\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135196794\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135196794\"]\n<p id=\"fs-id1165135196796\">[latex]{f}^{-1}\\left(x\\right)=x-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634366\">\n<div id=\"fs-id1165137634368\">\n<p id=\"fs-id1165137679711\">[latex]f\\left(x\\right)=x+5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706135\">\n<div id=\"fs-id1165137422592\">\n<p id=\"fs-id1165137422594\">[latex]f\\left(x\\right)=2-x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137812372\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137812372\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137812372\"]\n<p id=\"fs-id1165137812374\">[latex]{f}^{-1}\\left(x\\right)=2-x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137561652\">\n<div id=\"fs-id1165137653456\">\n<p id=\"fs-id1165137653458\">[latex]f\\left(x\\right)=3-x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600416\">\n<div id=\"fs-id1165135198605\">\n<p id=\"fs-id1165135198608\">[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135541959\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135541959\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135541959\"]\n<p id=\"fs-id1165135541961\">[latex]{f}^{-1}\\left(x\\right)=\\frac{-2x}{x-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137541591\">\n<div>\n<p id=\"fs-id1165134043733\">[latex]f\\left(x\\right)=\\frac{2x+3}{5x+4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137410615\">For the following exercises, find a domain on which each function[latex]\\,f\\,[\/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\\,f\\,[\/latex]restricted to that domain.<\/p>\n\n<div id=\"fs-id1165134148519\">\n<div id=\"fs-id1165134148521\">\n<p id=\"fs-id1165137737103\">[latex]f\\left(x\\right)={\\left(x+7\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137849508\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137849508\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137849508\"]\n<p id=\"fs-id1165137655378\">domain of[latex]f\\left(x\\right):\\,\\left[-7,\\infty \\right);\\,{f}^{-1}\\left(x\\right)=\\sqrt{x}-7[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862703\">\n<div id=\"fs-id1165137531119\">\n<p id=\"fs-id1165137531121\">[latex]f\\left(x\\right)={\\left(x-6\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137938700\">\n<p id=\"fs-id1165134042196\">[latex]f\\left(x\\right)={x}^{2}-5[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137603366\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137603366\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137603366\"]\n<p id=\"fs-id1165137603368\">domain of[latex]\\,f\\left(x\\right):\\,\\left[0,\\infty \\right);\\,{f}^{-1}\\left(x\\right)=\\sqrt{x+5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734966\">\n<div id=\"fs-id1165137734969\">\n<p id=\"fs-id1165137731914\">Given[latex]\\,f\\left(x\\right)=\\frac{x}{2+x}\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\frac{2x}{1-x}:[\/latex]<\/p>\n\n<ol id=\"fs-id1165137838751\" type=\"a\">\n \t<li>Find[latex]\\,f\\left(g\\left(x\\right)\\right)\\,[\/latex]and[latex]\\,g\\left(f\\left(x\\right)\\right).[\/latex]<\/li>\n \t<li>What does the answer tell us about the relationship between[latex]\\,f\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137730082\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137730082\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137730082\"]\n<p id=\"fs-id1165137730084\">a.[latex] \\,f\\left(g\\left(x\\right)\\right)=x\\,[\/latex]and[latex]\\,g\\left(f\\left(x\\right)\\right)=x.\\,[\/latex]b. This tells us that[latex]\\,f\\,[\/latex]and[latex]\\,g\\,[\/latex]are inverse functions<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137444427\">For the following exercises, use function composition to verify that[latex]\\,f\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)\\,[\/latex]are inverse functions.<\/p>\n\n<div id=\"fs-id1165137437578\">\n<div id=\"fs-id1165137619341\">\n<p id=\"fs-id1165137619343\">[latex]f\\left(x\\right)=\\sqrt[3]{x-1}\\,[\/latex]and[latex]\\,g\\left(x\\right)={x}^{3}+1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137883792\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137883792\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137883792\"]\n<p id=\"fs-id1165135310692\">[latex] f\\left(g\\left(x\\right)\\right)=x,\\,g\\left(f\\left(x\\right)\\right)=x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538749\">\n<div id=\"fs-id1165135538751\">\n<p id=\"fs-id1165137452674\">[latex]f\\left(x\\right)=-3x+5\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\frac{x-5}{-3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188614\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135176520\">For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n\n<div id=\"fs-id1165137645254\">\n<div>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/div>\n<div id=\"fs-id1165134113903\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134113903\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134113903\"]\n<p id=\"fs-id1165137534912\">one-to-one<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173420\">\n<div id=\"fs-id1165135173423\">\n<p id=\"fs-id1165135256110\">[latex]f\\left(x\\right)=\\sqrt[3]{3x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135424683\">\n<div id=\"fs-id1165137408415\">\n<p id=\"fs-id1165137408417\">[latex]f\\left(x\\right)=-5x+1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137433244\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137433244\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137433244\"]\n<p id=\"fs-id1165137393282\">one-to-one<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704606\">\n<div id=\"fs-id1165137704608\">\n<p id=\"fs-id1165137837047\">[latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137528556\">For the following exercises, determine whether the graph represents a one-to-one function.<\/p>\n\n<div id=\"fs-id1165137528559\">\n<div id=\"fs-id1165135251340\"><span id=\"fs-id1165135341390\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142115\/CNX_Precalc_Figure_01_07_201.jpg\" alt=\"Graph of a parabola.\"><\/span><\/div>\n<div id=\"fs-id1165137612243\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137612243\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137612243\"]\n<p id=\"fs-id1165131959461\">not one-to-one<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192971\">\n<div id=\"fs-id1165135192973\"><span id=\"fs-id1165134379457\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142130\/CNX_Precalc_Figure_01_07_202.jpg\" alt=\"Graph of a step-function.\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165137849556\">For the following exercises, use the graph of[latex]\\,f\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_203\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_203\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137855139\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142133\/CNX_Precalc_Figure_01_07_203.jpg\" alt=\"Graph of a line.\"><\/span><\/div>\n<div id=\"fs-id1165137863913\">\n<div id=\"fs-id1165137854842\">\n<p id=\"fs-id1165137854844\">Find[latex]\\,f\\left(0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137653684\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137653684\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137653684\"]\n<p id=\"fs-id1165137851374\">[latex]3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417814\">\n<div id=\"fs-id1165137417816\">\n<p id=\"fs-id1165137417818\">Solve[latex]\\,f\\left(x\\right)=0.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133093360\">\n<div id=\"fs-id1165133093363\">\n<p id=\"fs-id1165137461116\">Find[latex]\\,{f}^{-1}\\left(0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137573270\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137573270\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137573270\"]\n<p id=\"fs-id1165137573273\">[latex]2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451393\">\n<div id=\"fs-id1165137806036\">\n<p id=\"fs-id1165137806038\">Solve[latex]\\,{f}^{-1}\\left(x\\right)=0.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135639868\">For the following exercises, use the graph of the one-to-one function shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_204\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_204\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137611817\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142135\/CNX_Precalc_Figure_01_07_204.jpg\" alt=\"Graph of a square root function.\"><\/span><\/div>\n<div id=\"fs-id1165137884386\">\n<div id=\"fs-id1165137812123\">\n<p id=\"fs-id1165137812125\">Sketch the graph of[latex]\\,{f}^{-1}.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134113897\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134113897\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134113897\"]<span id=\"fs-id1165137939427\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142137\/CNX_Precalc_Figure_01_07_205.jpg\" alt=\"Graph of a square root function and its inverse.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137892244\">\n<div id=\"fs-id1165137603262\">\n<p id=\"fs-id1165137603264\">Find[latex]\\,f\\left(6\\right)\\text{ and }{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827763\">\n<div id=\"fs-id1165137827765\">\n<p id=\"fs-id1165137827767\">If the complete graph of[latex]\\,f\\,[\/latex]is shown, find the domain of[latex]\\,f.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134259264\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134259264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134259264\"]\n<p id=\"fs-id1165135699154\">[latex]\\left[2,10\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137644584\">\n<p id=\"fs-id1165135307895\">If the complete graph of[latex]\\,f\\,[\/latex]is shown, find the range of[latex]\\,f.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135434734\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137781583\">For the following exercises, evaluate or solve, assuming that the function[latex]\\,f\\,[\/latex]is one-to-one.<\/p>\n\n<div id=\"fs-id1165137771148\">\n<div id=\"fs-id1165135250622\">\n<p id=\"fs-id1165135250624\">If[latex]\\,f\\left(6\\right)=7,\\,[\/latex]find[latex]\\,\\,{f}^{-1}\\left(7\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137714192\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137714192\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137714192\"]\n<p id=\"fs-id1165137714194\">[latex]6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805757\">\n<div id=\"fs-id1165135190808\">\n<p id=\"fs-id1165135190810\">If[latex]\\,f\\left(3\\right)=2,\\,[\/latex]find[latex]\\,{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640685\">\n<div id=\"fs-id1165137640687\">\n<p id=\"fs-id1165137640689\">If[latex]\\,{f}^{-1}\\left(-4\\right)=-8,\\,[\/latex]find[latex]\\,f\\left(-8\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137848946\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137848946\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137848946\"]\n<p id=\"fs-id1165135169413\">[latex]-4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404972\">\n<div id=\"fs-id1165137404974\">\n<p id=\"fs-id1165137404976\">If[latex]\\,{f}^{-1}\\left(-2\\right)=-1,\\,[\/latex]find[latex]\\,f\\left(-1\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135195398\">For the following exercises, use the values listed in <a class=\"autogenerated-content\" href=\"#Table_01_07_06\">(Figure)<\/a> to evaluate or solve.<\/p>\n\n<table id=\"Table_01_07_06\" summary=\"Two columns and ten rows. The first column is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So for f(0)=8, f(1)=0, f(2)=7, f(3)=4, f(4)=2, f(5)=6, f(6)=5, f(7)=8, f(8)=9, and f(9)=1.\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137400581\">\n<div id=\"fs-id1165137400584\">\n<p id=\"fs-id1165137400586\">Find[latex]\\,f\\left(1\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135263655\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135263655\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135263655\"]\n<p id=\"fs-id1165137823491\">[latex]0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137807065\">\n<div id=\"fs-id1165137807068\">\n<p id=\"fs-id1165137645884\">Solve[latex]\\,f\\left(x\\right)=3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736914\">\n<div id=\"fs-id1165137736916\">\n<p id=\"fs-id1165137871008\">Find[latex]\\,{f}^{-1}\\left(0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137422471\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137422471\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137422471\"]\n<p id=\"fs-id1165137422474\">[latex]\\,1\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137686729\">\n<div id=\"fs-id1165137686731\">\n<p id=\"fs-id1165137686733\">Solve[latex]\\,{f}^{-1}\\left(x\\right)=7.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135209686\">\n<p id=\"fs-id1165135209688\">Use the tabular representation of[latex]\\,f\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Table_01_07_08\">(Figure)<\/a> to create a table for[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n\n<table id=\"Table_01_07_08\" summary=\"Two rows and six columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 3, 6, 9, 13, and 14. So for f(3)=1, f(6)=4, f(9)=7, f(13)=12, and f(14)=16.\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>3<\/td>\n<td>6<\/td>\n<td>9<\/td>\n<td>13<\/td>\n<td>14<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>4<\/td>\n<td>7<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137406963\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137406963\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137406963\"]\n<table id=\"fs-id1165134138595\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf^(-1)(x)\u201d. The values of x are 1, 4, 7, 12, and 16. So for f^(-1) (1)=1, f^(-1) (4)=6, f^(-1) (7)=9, f^(-1) (12)=13, and f^(-1)f(16)=14.\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>1<\/td>\n<td>4<\/td>\n<td>7<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>[latex]{f}^{-1}\\left(x\\right)[\/latex]<\/td>\n<td>3<\/td>\n<td>6<\/td>\n<td>9<\/td>\n<td>13<\/td>\n<td>14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641552\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137401884\">For the following exercises, find the inverse function. Then, graph the function and its inverse.<\/p>\n\n<div id=\"fs-id1165137401888\">\n<div id=\"fs-id1165137470994\">\n<p id=\"fs-id1165137470996\">[latex]f\\left(x\\right)=\\frac{3}{x-2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135435667\">\n<div id=\"fs-id1165135435669\">\n<p id=\"fs-id1165135208995\">[latex]f\\left(x\\right)={x}^{3}-1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135541804\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135541804\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135541804\"]\n<p id=\"fs-id1165135541807\">[latex]{f}^{-1}\\left(x\\right)={\\left(1+x\\right)}^{1\/3}[\/latex]<\/p>\n<span id=\"fs-id1165134057534\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142141\/CNX_Precalc_Figure_01_07_207.jpg\" alt=\"Graph of a cubic function and its inverse.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419353\">\n<div>\n<p id=\"fs-id1165135209711\">Find the inverse function of[latex]\\,f\\left(x\\right)=\\frac{1}{x-1}.\\,[\/latex]Use a graphing utility to find its domain and range. Write the domain and range in interval notation.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137592239\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135512365\">\n<div id=\"fs-id1165135512368\">\n<p id=\"fs-id1165135194429\">To convert from[latex]\\,x\\,[\/latex]degrees Celsius to[latex]\\,y\\,[\/latex]degrees Fahrenheit, we use the formula[latex]\\,f\\left(x\\right)=\\frac{9}{5}x+32.\\,[\/latex]Find the inverse function, if it exists, and explain its meaning.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134164967\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134164967\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134164967\"]\n<p id=\"fs-id1165134164969\">[latex]{f}^{-1}\\left(x\\right)=\\frac{5}{9}\\left(x-32\\right).\\,[\/latex]Given the Fahrenheit temperature,[latex]\\,x,\\,[\/latex]this formula allows you to calculate the Celsius temperature.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462831\">\n<div id=\"fs-id1165137462833\">\n<p id=\"fs-id1165137462835\">The circumference[latex]\\,C\\,[\/latex]of a circle is a function of its radius given by[latex]\\,C\\left(r\\right)=2\\pi r.\\,[\/latex]Express the radius of a circle as a function of its circumference. Call this function[latex]\\,r\\left(C\\right).\\,[\/latex]Find[latex]\\,r\\left(36\\pi \\right)\\,[\/latex]and interpret its meaning.<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135532476\">\n<p id=\"fs-id1165137645594\">A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time,[latex]\\,t,\\,[\/latex]in hours given by[latex]\\,d\\left(t\\right)=50t.\\,[\/latex]Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function[latex]\\,t\\left(d\\right).\\,[\/latex]Find[latex]\\,t\\left(180\\right)\\,[\/latex]and interpret its meaning.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137766909\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137766909\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137766909\"]\n<p id=\"fs-id1165137766912\">[latex]t\\left(d\\right)=\\frac{d}{50},\\,[\/latex][latex]t\\left(180\\right)=\\frac{180}{50}.\\,[\/latex]The time for the car to travel 180 miles is 3.6 hours.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135177582\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"eip-id1165135176875\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/55f2e8ec-a982-4586-9d48-a2f43d7b4107\">Functions and Function Notation<\/a><\/h4>\n<p id=\"fs-id1165137911358\">For the following exercises, determine whether the relation is a function.<\/p>\n\n<div id=\"fs-id1165137464074\">\n<div>[latex]\\left\\{\\left(a,b\\right),\\left(c,d\\right),\\left(e,d\\right)\\right\\}[\/latex]<\/div>\n<div id=\"fs-id1165137552980\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137552980\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137552980\"]\n<p id=\"fs-id1165137552982\">function<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137728083\">\n<div id=\"fs-id1165137501972\">[latex]\\left\\{\\left(5,2\\right),\\left(6,1\\right),\\left(6,2\\right),\\left(4,8\\right)\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134569510\">\n<div id=\"fs-id1165134569512\">\n<p id=\"fs-id1165137446921\">[latex]{y}^{2}+4=x,\\,[\/latex]for[latex]\\,x\\,[\/latex]the independent variable and[latex]\\,y\\,[\/latex]the dependent variable<\/p>\n\n<\/div>\n<div id=\"fs-id1165134081333\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134081333\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134081333\"]\n<p id=\"fs-id1165134081336\">not a function<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137464340\">\n<div id=\"fs-id1165137464343\">\n<p id=\"fs-id1165137828359\">Is the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_07_208\">(Figure)<\/a> a function?<\/p>\n\n<div id=\"Figure_01_07_208\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165135154408\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142202\/CNX_Precalc_Figure_01_07_208.jpg\" alt=\"Graph of a parabola.\"><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137417004\">For the following exercises, evaluate the function at the indicated values:[latex]\\,\\,\\,f\\left(-3\\right);\\,\\,f\\left(2\\right);\\,\\,\\,f\\left(-a\\right);\\,\\,\\,-f\\left(a\\right);\\,\\,\\,f\\left(a+h\\right).[\/latex]<\/p>\n\n<div id=\"fs-id1165135186568\">\n<div id=\"fs-id1165135186571\">\n<p id=\"fs-id1165137728372\">[latex]f\\left(x\\right)=-2{x}^{2}+3x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137401744\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137401744\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137401744\"]\n<p id=\"fs-id1165137446132\">[latex]f\\left(-3\\right)=-27;[\/latex][latex]f\\left(2\\right)=-2;[\/latex][latex]f\\left(-a\\right)=-2{a}^{2}-3a;[\/latex]<\/p>\n[latex]-f\\left(a\\right)=2{a}^{2}-3a;[\/latex][latex]f\\left(a+h\\right)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}[\/latex][\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165137651645\">\n<div id=\"fs-id1165137651647\">\n<p id=\"fs-id1165137651649\">[latex]f\\left(x\\right)=2|3x-1|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135496646\">For the following exercises, determine whether the functions are one-to-one.<\/p>\n\n<div id=\"fs-id1165137605472\">\n<div id=\"fs-id1165137605474\">\n<p id=\"fs-id1165137580535\">[latex]f\\left(x\\right)=-3x+5[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135181580\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135181580\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135181580\"]\n<p id=\"fs-id1165135181582\">one-to-one<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731102\">\n<div id=\"fs-id1165137643347\">\n<p id=\"fs-id1165137643349\">[latex]f\\left(x\\right)=|x-3|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137541296\">For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.<\/p>\n\n<div id=\"fs-id1165134324955\">\n<div id=\"fs-id1165134324957\"><span id=\"fs-id1165137416484\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142209\/CNX_Precalc_Figure_01_07_209.jpg\" alt=\"Graph of a cubic function.\"><\/span><\/div>\n<div id=\"fs-id1165137784950\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137784950\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137784950\"]\n<p id=\"fs-id1165137784952\">function<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541845\">\n<div id=\"fs-id1165135541847\"><span id=\"fs-id1165137612052\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142211\/CNX_Precalc_Figure_01_07_210.jpg\" alt=\"Graph of a relation.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137653012\">\n<div id=\"fs-id1165137427968\"><span id=\"fs-id1165135168455\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142213\/CNX_Precalc_Figure_01_07_211.jpg\" alt=\"Graph of a relation.\"><\/span><\/div>\n<div id=\"fs-id1165137626904\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137626904\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137626904\"]\n<p id=\"fs-id1165137626907\">function<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135255816\">For the following exercises, graph the functions.<\/p>\n\n<div id=\"fs-id1165137936642\">\n<div id=\"fs-id1165137936644\">\n<p id=\"fs-id1165137936646\">[latex]f\\left(x\\right)=|x+1|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137454708\">\n<div id=\"fs-id1165137454711\">\n<p id=\"fs-id1165137454713\">[latex]f\\left(x\\right)={x}^{2}-2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137441682\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137441682\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137441682\"]<span id=\"fs-id1165137641445\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142215\/CNX_Precalc_Figure_01_07_213.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"eip-860\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_01_07_215\">(Figure)<\/a> to approximate the values.<\/p>\n\n<div id=\"Figure_01_07_215\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137536184\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142225\/CNX_Precalc_Figure_01_07_215.jpg\" alt=\"Graph of a parabola.\"><\/span><\/div>\n<div id=\"fs-id1165137603626\">\n<div id=\"fs-id1165137603628\">\n<p id=\"fs-id1165135255923\">[latex]f\\left(2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134257596\">\n<div id=\"fs-id1165137579612\">\n<p id=\"fs-id1165137579614\">[latex]f\\left(-2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134269533\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134269533\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134269533\"]\n<p id=\"fs-id1165134269535\">[latex]2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722552\">\n<div id=\"fs-id1165137722554\">\n<p id=\"fs-id1165137722556\">If[latex]\\,f\\left(x\\right)=-2,\\,[\/latex]then solve for[latex]\\,x.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823750\">\n<div id=\"fs-id1165137456896\">\n<p id=\"fs-id1165137456898\">If[latex]\\,f\\left(x\\right)=1,\\,[\/latex]then solve for[latex]\\,x.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137871525\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137871525\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137871525\"]\n<p id=\"fs-id1165137462412\">[latex]x=-1.8\\text{ }[\/latex]or[latex]\\text{ or }x=1.8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137642827\">For the following exercises, use the function[latex]\\,h\\left(t\\right)=-16{t}^{2}+80t\\,[\/latex]to find the values in simplest form.<\/p>\n\n<div id=\"fs-id1165135160654\">\n<div id=\"fs-id1165135160656\">\n<p id=\"fs-id1165135160658\">[latex]\\frac{h\\left(2\\right)-h\\left(1\\right)}{2-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742766\">\n<div id=\"fs-id1165137742768\">\n<p id=\"fs-id1165137742770\">[latex]\\frac{h\\left(a\\right)-h\\left(1\\right)}{a-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135409755\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135409755\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135409755\"]\n<p id=\"fs-id1165135409757\">[latex]\\frac{-64+80a-16{a}^{2}}{-1+a}=-16a+64[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165132944714\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/562c3737-a93d-458c-98c0-a04f442f13bd\">Domain and Range<\/a><\/h4>\n<p id=\"fs-id1165137667477\">For the following exercises, find the domain of each function, expressing answers using interval notation.<\/p>\n\n<div id=\"fs-id1165137667482\">\n<div id=\"fs-id1165137535162\">\n<p id=\"fs-id1165137535164\">[latex]f\\left(x\\right)=\\frac{2}{3x+2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648069\">\n<div id=\"fs-id1165137648071\">\n<p id=\"fs-id1165137936848\">[latex]f\\left(x\\right)=\\frac{x-3}{{x}^{2}-4x-12}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137552661\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137552661\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137552661\"]\n<p id=\"fs-id1165137552663\">[latex]\\left(-\\infty ,-2\\right)\\cup \\left(-2,6\\right)\\cup \\left(6,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137933220\">\n<div id=\"fs-id1165131990652\">\n<p id=\"fs-id1165131990654\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135368485\">\n<div id=\"fs-id1165137665507\">\n<p id=\"fs-id1165137665510\">Graph this piecewise function:[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{l}x+1\\text{ }x&lt;-2\\\\ -2x-3\\text{ }x\\ge -2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133437266\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133437266\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133437266\"]<span id=\"fs-id1165137803119\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142226\/CNX_Precalc_Figure_01_07_214.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165133183577\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f37919d7-b496-4e36-8196-431ae4733a64\">Rates of Change and Behavior of Graphs<\/a><\/h4>\n<p id=\"fs-id1165135161227\">For the following exercises, find the average rate of change of the functions from[latex]\\,x=1\\text{ to }x=2.[\/latex]<\/p>\n\n<div id=\"fs-id1165137541046\">\n<div id=\"fs-id1165137541048\">\n<p id=\"fs-id1165137541051\">[latex]f\\left(x\\right)=4x-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137664379\">\n<div id=\"fs-id1165137664381\">\n<p id=\"fs-id1165137664383\">[latex]f\\left(x\\right)=10{x}^{2}+x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134148505\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134148505\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134148505\"]\n<p id=\"fs-id1165134148508\">[latex]31[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137807042\">\n<div id=\"fs-id1165137807044\">\n<p id=\"fs-id1165137601405\">[latex]f\\left(x\\right)=-\\frac{2}{{x}^{2}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137863445\">For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.<\/p>\n\n<div id=\"fs-id1165137527205\">\n<div id=\"fs-id1165137527207\"><span id=\"fs-id1165137734838\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142228\/CNX_Precalc_Figure_01_07_216.jpg\" alt=\"Graph of a parabola.\"><\/span><\/div>\n<div id=\"fs-id1165137911127\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137911127\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137911127\"]\n<p id=\"fs-id1165137911128\">increasing[latex]\\,\\left(2,\\infty \\right);\\,[\/latex]\ndecreasing[latex]\\,\\left(-\\infty ,2\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575929\">\n<div id=\"fs-id1165137768455\"><span id=\"fs-id1165135169395\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142240\/CNX_Precalc_Figure_01_07_217.jpg\" alt=\"Graph of a cubic function.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135153146\">\n<div id=\"fs-id1165135153148\"><span id=\"fs-id1165137640124\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142247\/CNX_Precalc_Figure_01_07_218.jpg\" alt=\"Graph of a function.\"><\/span><\/div>\n<div id=\"fs-id1165135439993\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135439993\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135439993\"]\n<p id=\"fs-id1165137810327\">increasing[latex]\\text{}\\left(-3,1\\right);\\text{}[\/latex]constant[latex]\\,\\left(-\\infty ,-3\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727192\">\n<div id=\"fs-id1165137727194\">\n<p id=\"fs-id1165137727197\">Find the local minimum of the function graphed in <a class=\"autogenerated-content\" href=\"#fs-id1165137527205\">(Figure)<\/a>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135183100\">\n<div id=\"fs-id1165135183102\">\n<p id=\"fs-id1165135259506\">Find the local extrema for the function graphed in <a class=\"autogenerated-content\" href=\"#fs-id1165137575929\">(Figure)<\/a>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135259511\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135259511\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135259511\"]\n<p id=\"fs-id1165137643970\">local minimum[latex]\\,\\left(-2,-3\\right);\\,[\/latex]local maximum[latex]\\,\\left(1,3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453948\">\n<div id=\"fs-id1165137453950\">\n<p id=\"fs-id1165137453952\">For the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_07_219\">(Figure)<\/a>, the domain of the function is[latex]\\,\\left[-3,3\\right].[\/latex]The range is[latex]\\,\\left[-10,10\\right].\\,[\/latex]Find the absolute minimum of the function on this interval.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575004\">\n<div id=\"fs-id1165137575006\">\n<p id=\"fs-id1165137597151\">Find the absolute maximum of the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_01_07_219\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_219\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165137441761\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142300\/CNX_Precalc_Figure_01_07_219.jpg\" alt=\"Graph of a cubic function.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137619711\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137619711\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137619711\"]\n<p id=\"fs-id1165137619713\">[latex]\\,\\left(-1.8,10\\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165131815810\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/8a4fc477-43da-4fc4-9ff0-19a7fae5a19d\">Composition of Functions<\/a><\/h4>\n<p id=\"fs-id1165132962099\">For the following exercises, find[latex]\\,\\left(f\\circ g\\right)\\left(x\\right)\\,[\/latex]and[latex]\\,\\left(g\\circ f\\right)\\left(x\\right)\\,[\/latex]for each pair of functions.<\/p>\n\n<div id=\"fs-id1165137535286\">\n<div id=\"fs-id1165137535288\">\n<p id=\"fs-id1165137653345\">[latex]f\\left(x\\right)=4-x,\\,g\\left(x\\right)=-4x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137581604\">\n<div id=\"fs-id1165137581606\">\n<p id=\"fs-id1165137581608\">[latex]f\\left(x\\right)=3x+2,\\,g\\left(x\\right)=5-6x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137401055\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137401055\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137401055\"]\n<p id=\"fs-id1165137401058\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=17-18x;\\,\\left(g\\circ f\\right)\\left(x\\right)=-7-18x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556913\">\n<div id=\"fs-id1165137556915\">\n<p id=\"fs-id1165137556917\">[latex]f\\left(x\\right)={x}^{2}+2x,\\,g\\left(x\\right)=5x+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135508544\">\n<div id=\"fs-id1165135508547\">\n<p id=\"fs-id1165133337518\">[latex]f\\left(x\\right)=\\sqrt{x+2},\\text{ }g\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137506826\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137506826\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137506826\"]\n<p id=\"fs-id1165137506828\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\sqrt{\\frac{1}{x}+2};\\,\\left(g\\circ f\\right)\\left(x\\right)=\\frac{1}{\\sqrt{x+2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134505601\">\n<div id=\"fs-id1165137437882\">\n<p id=\"fs-id1165137437884\">[latex]\\,f\\left(x\\right)=\\frac{x+3}{2},\\text{ }g\\left(x\\right)=\\sqrt{1-x}\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137411081\">For the following exercises, find[latex]\\,\\left(f\\circ g\\right)\\,[\/latex]and the domain for[latex]\\,\\left(f\\circ g\\right)\\left(x\\right)\\,[\/latex]for each pair of functions.<\/p>\n\n<div id=\"fs-id1165137851236\">\n<div id=\"fs-id1165137851238\">\n<p id=\"fs-id1165137851240\">[latex]f\\left(x\\right)=\\frac{x+1}{x+4},\\text{ }g\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137472983\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137472983\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137472983\"]\n<p id=\"fs-id1165137588201\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\frac{1+x}{1+4x}, x\\ne 0, x\\ne -\\frac{1}{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407511\">\n<div id=\"fs-id1165135407513\">\n<p id=\"fs-id1165135407515\">[latex]f\\left(x\\right)=\\frac{1}{x+3},\\text{ }g\\left(x\\right)=\\frac{1}{x-9}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438445\">\n<div id=\"fs-id1165137596259\">\n<p id=\"fs-id1165137596261\">[latex]f\\left(x\\right)=\\frac{1}{x},\\text{ }g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134374025\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134374025\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134374025\"]\n<p id=\"fs-id1165137456076\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\frac{1}{\\sqrt{x}},\\,x&gt;0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135505012\">\n<div id=\"fs-id1165135505014\">\n<p id=\"fs-id1165137427123\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-1},\\text{ }g\\left(x\\right)=\\sqrt{x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137734804\">For the following exercises, express each function[latex]\\,H\\,[\/latex]as a composition of two functions[latex]\\,f\\,[\/latex]and[latex]\\,g\\,[\/latex]where[latex]\\,H\\left(x\\right)=\\left(f\\circ g\\right)\\left(x\\right).[\/latex]<\/p>\n\n<div id=\"fs-id1165137448277\">\n<div id=\"fs-id1165137448279\">\n<p id=\"fs-id1165137837225\">[latex]H\\left(x\\right)=\\sqrt{\\frac{2x-1}{3x+4}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137697087\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137697087\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137697087\"]\n<p id=\"fs-id1165137697090\">sample:[latex]\\,g\\left(x\\right)=\\frac{2x-1}{3x+4};\\,f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526679\">\n<div id=\"fs-id1165137526681\">\n<p id=\"fs-id1165137526683\">[latex]H\\left(x\\right)=\\frac{1}{{\\left(3{x}^{2}-4\\right)}^{-3}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165134070725\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/5f6ff02a-1000-410d-b034-af26fbd86d0b\">Transformation of Functions<\/a><\/h4>\n<p id=\"fs-id1165137901175\">For the following exercises, sketch a graph of the given function.<\/p>\n\n<div id=\"fs-id1165137901178\">\n<div id=\"fs-id1165135439826\">\n<p id=\"fs-id1165135439828\">[latex]f\\left(x\\right)={\\left(x-3\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137665486\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137665486\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137665486\"]<span id=\"fs-id1165135416583\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142314\/CNX_Precalc_Figure_01_07_220.jpg\" alt=\"Graph of f(x)\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137419235\">\n<div id=\"fs-id1165137419237\">\n<p id=\"fs-id1165135358013\">[latex]f\\left(x\\right)={\\left(x+4\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605818\">\n<div id=\"fs-id1165137597708\">\n<p id=\"fs-id1165137597710\">[latex]f\\left(x\\right)=\\sqrt{x}+5[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137442696\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137442696\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137442696\"]<span id=\"fs-id1165137768206\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142316\/CNX_Precalc_Figure_01_07_222.jpg\" alt=\"Graph of f(x)\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137638654\">\n<div id=\"fs-id1165137638656\">\n<p id=\"fs-id1165134386603\">[latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137446722\">\n<div id=\"fs-id1165137565755\">\n<p id=\"fs-id1165137565757\">[latex]f\\left(x\\right)=\\sqrt[3]{-x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137771419\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137771419\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137771419\"]<span id=\"fs-id1165135255841\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142321\/CNX_Precalc_Figure_01_07_224.jpg\" alt=\"Graph of f(x)\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137619832\">\n<div id=\"fs-id1165135421539\">\n<p id=\"fs-id1165135421541\">[latex]f\\left(x\\right)=5\\sqrt{-x}-4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137793577\">\n<div id=\"fs-id1165137793579\">\n<p id=\"fs-id1165137793581\">[latex]f\\left(x\\right)=4\\left[|x-2|-6\\right][\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137651999\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137651999\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137651999\"]<span id=\"fs-id1165137599946\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142330\/CNX_Precalc_Figure_01_07_226.jpg\" alt=\"Graph of f(x)\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137628681\">\n<div id=\"fs-id1165137628683\">\n<p id=\"fs-id1165137628685\">[latex]f\\left(x\\right)=-{\\left(x+2\\right)}^{2}-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137706346\">For the following exercises, sketch the graph of the function[latex]\\,g\\,[\/latex]if the graph of the function[latex]\\,f\\,[\/latex]is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_247\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_247\" class=\"medium\"><span id=\"fs-id1165137460564\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142332\/CNX_Precalc_Figure_01_07_247.jpg\" alt=\"Graph of f(x)\"><\/span><\/div>\n<div id=\"fs-id1165137736169\">\n<div id=\"fs-id1165137736171\">\n<p id=\"fs-id1165135152244\">[latex]g\\left(x\\right)=f\\left(x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137415538\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137415538\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137415538\"]<span id=\"fs-id1165137442236\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142334\/CNX_Precalc_Figure_01_07_228.jpg\" alt=\"Graph of a half circle.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135543526\">\n<div id=\"fs-id1165137742527\">\n<p id=\"fs-id1165137742529\">[latex]g\\left(x\\right)=3f\\left(x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137626728\">For the following exercises, write the equation for the standard function represented by each of the graphs below.<\/p>\n\n<div id=\"fs-id1165137562807\">\n<div id=\"fs-id1165137562809\"><span id=\"fs-id1165137446970\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142336\/CNX_Precalc_Figure_01_07_230.jpg\" alt=\"Graph of an absolute function.\"><\/span><\/div>\n<div id=\"fs-id1165137761321\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137761321\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137761321\"]\n<p id=\"fs-id1165137937561\">[latex]f\\left(x\\right)=|x-3|[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137570135\">\n<div id=\"fs-id1165137570137\"><span id=\"fs-id1165135524458\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142347\/CNX_Precalc_Figure_01_07_231.jpg\" alt=\"Graph of a half circle.\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165135496278\">For the following exercises, determine whether each function below is even, odd, or neither.<\/p>\n\n<div id=\"fs-id1165135496282\">\n<div id=\"fs-id1165137416232\">\n<p id=\"fs-id1165137416234\">[latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137812584\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137812584\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137812584\"]\n<p id=\"fs-id1165137812586\">even<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137394734\">\n<div id=\"fs-id1165137394736\">\n<p id=\"fs-id1165134258620\">[latex]g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176486\">\n<div id=\"fs-id1165135176488\">\n<p id=\"fs-id1165135176491\">[latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137598143\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137598143\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137598143\"]\n<p id=\"fs-id1165137598145\">odd<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137565963\">For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.<\/p>\n\n<div id=\"fs-id1165137565968\">\n<div id=\"fs-id1165134148336\"><span id=\"fs-id1165137534178\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142354\/CNX_Precalc_Figure_01_07_232.jpg\" alt=\"Graph of a parabola.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137643328\">\n<div id=\"fs-id1165137643331\"><span id=\"fs-id1165137836434\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142402\/CNX_Precalc_Figure_01_07_233.jpg\" alt=\"Graph of a parabola.\"><\/span><\/div>\n<div id=\"fs-id1165135344925\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135344925\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135344925\"]\n<p id=\"fs-id1165135545915\">even<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135471211\">\n<div id=\"fs-id1165135471213\"><span id=\"fs-id1165135538487\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142408\/CNX_Precalc_Figure_01_07_234.jpg\" alt=\"Graph of a cubic function.\"><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165135691363\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/2e387575-c04f-40e1-8895-195affae8fdb\">Absolute Value Functions<\/a><\/h4>\n<p id=\"fs-id1165135203710\">For the following exercises, write an equation for the transformation of[latex]\\,f\\left(x\\right)=|x|.[\/latex]<\/p>\n\n<div id=\"fs-id1165137725549\">\n<div id=\"fs-id1165137470719\"><span id=\"fs-id1165137597393\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142411\/CNX_Precalc_Figure_01_07_235.jpg\" alt=\"Graph of f(x).\"><\/span><\/div>\n<div id=\"fs-id1165137635245\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137635245\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137635245\"]\n<p id=\"fs-id1165137635247\">[latex]f\\left(x\\right)=\\frac{1}{2}|x+2|+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836804\">\n<div id=\"fs-id1165137836807\"><span id=\"fs-id1165137849182\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142412\/CNX_Precalc_Figure_01_07_236.jpg\" alt=\"Graph of f(x).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137737661\">\n<div id=\"fs-id1165134224080\"><span id=\"fs-id1165137656022\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142413\/CNX_Precalc_Figure_01_07_237.jpg\" alt=\"Graph of f(x).\"><\/span><\/div>\n<div id=\"fs-id1165135181351\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135181351\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135181351\"]\n<p id=\"fs-id1165135181354\">[latex]f\\left(x\\right)=-3|x-3|+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137415922\">For the following exercises, graph the absolute value function.<\/p>\n\n<div id=\"fs-id1165137426335\">\n<div id=\"fs-id1165137426337\">\n<p id=\"fs-id1165137426339\">[latex]f\\left(x\\right)=|x-5|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134058433\">\n<div id=\"fs-id1165134058435\">\n<p id=\"fs-id1165134058437\">[latex]f\\left(x\\right)=-|x-3|[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134151182\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134151182\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134151182\"]<span id=\"fs-id1165135256168\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142421\/CNX_Precalc_Figure_01_07_239.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137673910\">\n<div id=\"fs-id1165135708035\">\n<p id=\"fs-id1165135708037\">[latex]f\\left(x\\right)=|2x-4|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-159\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f592aad0-19d8-42d6-94b9-086bdd84c2b5\">Inverse Functions<\/a><\/h4>\n<p id=\"eip-id1964739\">For the following exercises, find[latex]\\text{ }{f}^{-1}\\left(x\\right)\\text{ }[\/latex]for each function.<\/p>\n\n<div>\n<div id=\"fs-id1165135255940\">\n<p id=\"fs-id1165135255942\">[latex]f\\left(x\\right)=9+10x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"eip-553\">\n<div id=\"eip-75\">\n<p id=\"eip-735\">[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"eip-813\">\n<div class=\"textbox shaded\">[reveal-answer q=\"633714\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"633714\"][latex]{f}^{-1}\\left(x\\right)=\\frac{-2x}{x-1}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-293\">For the following exercise, find a domain on which the function[latex]\\text{ }f\\text{ }[\/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\\text{ }f\\text{ }[\/latex]restricted to that domain.<\/p>\n\n<div id=\"eip-117\">\n<div id=\"eip-537\">\n<p id=\"eip-385\">[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"eip-98\">\n<div id=\"eip-578\">\n<p id=\"eip-395\">Given [latex]f\\left(x\\right)={x}^{3}-5[\/latex] and [latex]g\\left(x\\right)=\\sqrt[3]{x+5}:[\/latex]<\/p>\n\n<ol id=\"eip-id1165134205830\" type=\"a\">\n \t<li>Find [latex] f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right).[\/latex]<\/li>\n \t<li>What does the answer tell us about the relationship between [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<div class=\"textbox shaded\">[reveal-answer q=\"730117\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"730117\"]\n<ol id=\"eip-id1165134350154\" type=\"a\">\n \t<li>[latex] f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x.[\/latex]<\/li>\n \t<li>This tells us that [latex]f[\/latex] and [latex]g[\/latex] are inverse functions<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-712\">For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n\n<div id=\"eip-72\">\n<div id=\"eip-808\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/div>\n<div id=\"eip-845\">\n<div class=\"textbox shaded\">[reveal-answer q=\"248056\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"248056\"]\n<p id=\"eip-366\">The function is one-to-one.<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142428\/CNX_Precalc_Figure_01_07_248.jpg\" alt=\"\">\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-642\">\n<div id=\"eip-85\">\n<p id=\"eip-256\">[latex]f\\left(x\\right)=-3{x}^{2}+x[\/latex]<\/p>\n\n<\/div>\n<div id=\"eip-id2171835\">\n<div class=\"textbox shaded\">[reveal-answer q=\"393417\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"393417\"]\n<p id=\"eip-id2171848\">The function is not one-to-one.<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142431\/CNX_Precalc_Figure_01_07_249.jpg\" alt=\"\">\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-491\">\n<div id=\"eip-368\">\n\nIf [latex]f\\left(5\\right)=2,[\/latex] find [latex]{f}^{-1}\\left(2\\right).[\/latex]\n<div class=\"textbox shaded\">\n<p id=\"eip-806\">[reveal-answer q=\"7234\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"7234\"]<\/p>\n\n<div id=\"eip-491\">\n<div id=\"eip-627\">\n<p id=\"eip-520\">[latex]5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-601\">\n<div id=\"eip-650\">\n<p id=\"eip-811\">If [latex]f\\left(1\\right)=4,[\/latex] find [latex]{f}^{-1}\\left(4\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137455205\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<p id=\"fs-id1165137590417\">For the following exercises, determine whether each of the following relations is a function.<\/p>\n\n<div id=\"fs-id1165137474063\">\n<div id=\"fs-id1165137474065\">\n<p id=\"fs-id1165137474067\">[latex]y=2x+8[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134572580\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134572580\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134572580\"]\n<p id=\"fs-id1165134572582\">The relation is a function.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736465\">\n<div id=\"fs-id1165137736467\">\n<p id=\"fs-id1165137832014\">[latex]\\left\\{\\left(2,1\\right),\\left(3,2\\right),\\left(-1,1\\right),\\left(0,-2\\right)\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137734462\">For the following exercises, evaluate the function[latex]\\,f\\left(x\\right)=-3{x}^{2}+2x\\,[\/latex]\nat the given input.<\/p>\n\n<div>\n<div>[latex]f\\left(-2\\right)[\/latex]<\/div>\n<div id=\"fs-id1165137472994\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137472994\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137472994\"]\n<p id=\"fs-id1165137472996\">\u221216<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135182930\">\n<div id=\"fs-id1165137535259\">\n<p id=\"fs-id1165137535261\">[latex]\\,f\\left(a\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541955\">\n<div id=\"fs-id1165137634239\">\n<p id=\"fs-id1165137634241\">Show that the function[latex]\\,f\\left(x\\right)=-2{\\left(x-1\\right)}^{2}+3\\,[\/latex]is not one-to-one.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135486043\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135486043\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135486043\"]\n<p id=\"fs-id1165135486045\">The graph is a parabola and the graph fails the horizontal line test.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573629\">\n<div id=\"fs-id1165137400957\">\n<p id=\"fs-id1165137400960\">Write the domain of the function[latex]\\,f\\left(x\\right)=\\sqrt{3-x}\\,[\/latex]in interval notation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137405680\">\n<div id=\"fs-id1165137405682\">\n<p id=\"fs-id1165135202594\">Given[latex]\\,f\\left(x\\right)=2{x}^{2}-5x,\\,[\/latex]find[latex]f\\left(a+1\\right)-f\\left(1\\right)\\,[\/latex]in simplest form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137455933\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137455933\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137455933\"]\n<p id=\"fs-id1165137812635\">[latex]2{a}^{2}-a[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137724050\">\n<div id=\"fs-id1165137724052\">\n<p id=\"fs-id1165137724054\">Graph the function[latex]f\\left(x\\right)=\\left\\{\\begin{array}{cc}x+1\\text{ if}&amp; -2&lt;x&lt;3\\\\ \\text{ }-x\\text{ if }&amp; x\\ge 3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810666\">\n<div id=\"fs-id1165137810669\">\n<p id=\"fs-id1165137810671\">Find the average rate of change of the function[latex]\\,f\\left(x\\right)=3-2{x}^{2}+x\\,[\/latex]by finding[latex]\\,\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}\\,[\/latex]in simplest form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135149027\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135149027\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135149027\"]\n<p id=\"fs-id1165135149029\">[latex]-2\\left(a+b\\right)+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137730928\">For the following exercises, use the functions[latex]\\,f\\left(x\\right)=3-2{x}^{2}+x\\text{ and }g\\left(x\\right)=\\sqrt{x}\\,[\/latex]to find the composite functions.<\/p>\n\n<div id=\"fs-id1165135332782\">\n<div id=\"fs-id1165137471389\">\n<p id=\"fs-id1165137471391\">[latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040544\">\n<div id=\"fs-id1165134040546\">\n<p id=\"fs-id1165134040549\">[latex]\\left(g\\circ f\\right)\\left(1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137572557\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137572557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137572557\"]\n<p id=\"fs-id1165137572559\">[latex]\\sqrt{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433228\">\n<div id=\"fs-id1165137433230\">\n<p id=\"fs-id1165137863995\">Express[latex]\\,H\\left(x\\right)=\\sqrt[3]{5{x}^{2}-3x}\\,[\/latex]as a composition of two functions,[latex]\\,f\\,[\/latex]and[latex]\\,g,\\,[\/latex]where[latex]\\,\\left(f\\circ g\\right)\\left(x\\right)=H\\left(x\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137849036\">For the following exercises, graph the functions by translating, stretching, and\/or compressing a toolkit function.<\/p>\n\n<div id=\"fs-id1165137724075\">\n<div id=\"fs-id1165137724077\">\n<p id=\"fs-id1165137714976\">[latex]f\\left(x\\right)=\\sqrt{x+6}-1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137400919\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137400919\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137400919\"]<span id=\"fs-id1165137454070\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142433\/CNX_Precalc_Figure_01_07_242.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137748677\">\n<div id=\"fs-id1165137748679\">\n<p id=\"fs-id1165137748681\">[latex]f\\left(x\\right)=\\frac{1}{x+2}-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135245866\">For the following exercises, determine whether the functions are even, odd, or neither.<\/p>\n\n<div id=\"fs-id1165137451069\">\n<div id=\"fs-id1165137451071\">\n<p id=\"fs-id1165137423842\">[latex]f\\left(x\\right)=-\\frac{5}{{x}^{2}}+9{x}^{6}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137610712\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137610712\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137610712\"]\n<p id=\"fs-id1165137610714\">[latex]\\text{even}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542465\">\n<div id=\"fs-id1165134250812\">[latex]f\\left(x\\right)=-\\frac{5}{{x}^{3}}+9{x}^{5}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137770368\">\n<div id=\"fs-id1165135536626\">\n<p id=\"fs-id1165135536628\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137540956\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137540956\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137540956\"]\n<p id=\"fs-id1165137534626\">[latex]\\text{odd}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137933901\">\n<div id=\"fs-id1165137933903\">\n<p id=\"fs-id1165137933906\">Graph the absolute value function[latex]\\,f\\left(x\\right)=-2|x-1|+3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137675389\">For the following exercises, find the inverse of the function.<\/p>\n\n<div id=\"fs-id1165137675392\">\n<div id=\"fs-id1165134389015\">\n<p id=\"fs-id1165134389017\">[latex]f\\left(x\\right)=3x-5[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135449688\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135449688\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135449688\"]\n<p id=\"fs-id1165135449690\">[latex]{f}^{-1}\\left(x\\right)=\\frac{x+5}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135195662\">\n<div id=\"fs-id1165135195664\">\n<p id=\"fs-id1165135195667\">[latex]f\\left(x\\right)=\\frac{4}{x+7}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137727655\">For the following exercises, use the graph of[latex]\\,g\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_245\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_245\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137456874\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142439\/CNX_Precalc_Figure_01_07_245.jpg\" alt=\"Graph of a cubic function.\"><\/span><\/div>\n<div id=\"fs-id1165137589849\">\n<div id=\"fs-id1165137589851\">\n<p id=\"fs-id1165137589853\">On what intervals is the function increasing?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137661790\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137661790\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137661790\"]\n<p id=\"fs-id1165137661792\">[latex]\\left(-\\infty ,-1.1\\right)\\text{ and }\\left(1.1,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137942391\">\n<div id=\"fs-id1165137942393\">\n<p id=\"fs-id1165137942395\">On what intervals is the function decreasing?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137463490\">\n<div id=\"fs-id1165134558010\">\n<p id=\"fs-id1165134558012\">Approximate the local minimum of the function. Express the answer as an ordered pair.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137619563\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137619563\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137619563\"]\n<p id=\"fs-id1165137619565\">[latex]\\left(1.1,-0.9\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135321930\">\n<div id=\"fs-id1165137679083\">\n<p id=\"fs-id1165137679085\">Approximate the local maximum of the function. Express the answer as an ordered pair.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137933195\">For the following exercises, use the graph of the piecewise function shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_246\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_07_246\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165135529085\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142444\/CNX_Precalc_Figure_01_07_246.jpg\" alt=\"Graph of absolute function and step function.\"><\/span><\/div>\n<div id=\"fs-id1165135517182\">\n<div id=\"fs-id1165135517184\">\n<p id=\"fs-id1165135517186\">Find[latex]\\,f\\left(2\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137762681\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137762681\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137762681\"]\n<p id=\"fs-id1165137762683\">[latex]f\\left(2\\right)=2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834540\">\n<div id=\"fs-id1165137443779\">\n<p id=\"fs-id1165137443781\">Find[latex]\\,f\\left(-2\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135168371\">\n<div id=\"fs-id1165135168373\">\n<p id=\"fs-id1165137556971\">Write an equation for the piecewise function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137556975\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137556975\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137556975\"]\n<p id=\"fs-id1165137726425\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}|x|\\,\\,\\,\\text{if}\\,\\,x\\le 2\\\\ 3\\,\\,\\,\\,\\,\\text{if}\\,\\,x&gt;2\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137696459\">For the following exercises, use the values listed in <a class=\"autogenerated-content\" href=\"#Table_01_07_07\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_01_07_07\" summary=\"..\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]F\\left(x\\right)[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>13<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165135149302\">\n<div id=\"fs-id1165135149304\">\n<p id=\"fs-id1165135149306\">Find[latex]\\,F\\left(6\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135616339\">\n<div id=\"fs-id1165135616341\">\n<p id=\"fs-id1165135616343\">Solve the equation[latex]\\,F\\left(x\\right)=5.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137676959\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137676959\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137676959\"]\n<p id=\"fs-id1165137676962\">[latex]x=2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836651\">\n<div id=\"fs-id1165137836653\">\n<p id=\"fs-id1165135256131\">Is the graph increasing or decreasing on its domain?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137850361\">\n<div id=\"fs-id1165137850363\">\n<p id=\"fs-id1165137850365\">Is the function represented by the graph one-to-one?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137736528\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137736528\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137736528\"]\n<p id=\"fs-id1165137736530\">yes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447422\">\n<div id=\"fs-id1165137447424\">\n<p id=\"fs-id1165137469729\">Find[latex]\\,{F}^{-1}\\left(15\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549524\">\n<div id=\"fs-id1165137549526\">\n<p id=\"fs-id1165137758292\">Given[latex]\\,f\\left(x\\right)=-2x+11,\\,[\/latex]find[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137851399\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137851399\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137851399\"]\n<p id=\"fs-id1165137851402\">[latex]{f}^{-1}\\left(x\\right)=-\\frac{x-11}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137441703\">\n \t<dt>inverse function<\/dt>\n \t<dd id=\"fs-id1165137441708\">for any one-to-one function[latex]\\,f\\left(x\\right),\\,[\/latex]the inverse is a function[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]such that[latex]\\,{f}^{-1}\\left(f\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f;\\,[\/latex]this also implies that[latex]\\,f\\left({f}^{-1}\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,{f}^{-1}[\/latex]<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Verify inverse functions.<\/li>\n<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.<\/li>\n<li>Find or evaluate the inverse of a function.<\/li>\n<li>Use the graph of a one-to-one function to graph its inverse function on the same axes.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135358875\">A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.<\/p>\n<p id=\"fs-id1165135701544\">If some physical machines can run in two directions, we might ask whether some of the function \u201cmachines\u201d we have been studying can also run backwards. <a class=\"autogenerated-content\" href=\"#Figure_01_07_001\">(Figure)<\/a> provides a visual representation of this question. In this section, we will consider the reverse nature of functions.<\/p>\n<div id=\"Figure_01_07_001\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141917\/CNX_Precalc_Figure_01_07_001.jpg\" alt=\"Diagram of a function and would be its inverse.\" width=\"731\" height=\"305\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Can a function \u201cmachine\u201d operate in reverse?<\/figcaption><\/figure>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<div id=\"fs-id1165137725994\" class=\"bc-section section\">\n<h3>Verifying That Two Functions Are Inverse Functions<\/h3>\n<p id=\"fs-id1165135705795\">Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the <span class=\"no-emphasis\">Celsius<\/span> scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees <span class=\"no-emphasis\">Fahrenheit<\/span> to degrees Celsius. She finds the formula<\/p>\n<div id=\"fs-id1165137807176\" class=\"unnumbered aligncenter\">[latex]C=\\frac{5}{9}\\left(F-32\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135433486\">and substitutes 75 for[latex]\\,F\\,[\/latex]to calculate<\/p>\n<div id=\"fs-id1165137911210\" class=\"unnumbered aligncenter\">[latex]\\frac{5}{9}\\left(75-32\\right)\\approx 24\\text{\u00b0C}[\/latex]<\/div>\n<p id=\"fs-id1165137409312\">Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week\u2019s weather forecast from <a class=\"autogenerated-content\" href=\"#Figure_01_07_002\">(Figure)<\/a> for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.<\/p>\n<div id=\"Figure_01_07_002\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141923\/CNX_Precalc_Figure_01_07_002.jpg\" alt=\"A forecast of Monday\u2019s through Thursday\u2019s weather.\" width=\"731\" height=\"226\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p>At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for[latex]\\,F\\,[\/latex]after substituting a value for[latex]\\,C.\\,[\/latex]For example, to convert 26 degrees Celsius, she could write<\/p>\n<div id=\"fs-id1165135548255\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 26& =& \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\ \\hfill 26\\cdot \\frac{9}{5}& =& F-32\\hfill \\\\ \\hfill F& =& 26\\cdot \\frac{9}{5}+32\\approx 79\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137540705\">After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.<\/p>\n<p id=\"fs-id1165137827441\">The formula for which Betty is searching corresponds to the idea of an <strong>inverse function<\/strong>, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.<\/p>\n<p id=\"fs-id1165135528385\">Given a function[latex]\\,f\\left(x\\right),\\,[\/latex]we represent its inverse as[latex]\\,{f}^{-1}\\left(x\\right),\\,[\/latex]read as[latex]\\,\u201cf\\,[\/latex]inverse of[latex]\\,x.\\text{\u201d}\\,[\/latex]The raised[latex]\\,-1\\,[\/latex]is part of the notation. It is not an exponent; it does not imply a power of[latex]\\,-1\\,[\/latex]. In other words,[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]does <em>not<\/em> mean[latex]\\,\\frac{1}{f\\left(x\\right)}\\,[\/latex]because[latex]\\,\\frac{1}{f\\left(x\\right)}\\,[\/latex]is the reciprocal of[latex]\\,f\\,[\/latex]and not the inverse.<\/p>\n<p id=\"fs-id1165137724926\">The \u201cexponent-like\u201d notation comes from an analogy between function composition and multiplication: just as[latex]\\,{a}^{-1}a=1\\,[\/latex](1 is the identity element for multiplication) for any nonzero number[latex]\\,a,\\,[\/latex]so[latex]\\,{f}^{-1}\\circ f\\,[\/latex]equals the identity function, that is,<\/p>\n<div id=\"fs-id1165134302408\" class=\"unnumbered aligncenter\">[latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(f\\left(x\\right)\\right)={f}^{-1}\\left(y\\right)=x[\/latex]<\/div>\n<p id=\"fs-id1165135667832\">This holds for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f.\\,[\/latex]Informally, this means that inverse functions \u201cundo\u201d each other. However, just as zero does not have a <span class=\"no-emphasis\">reciprocal<\/span>, some functions do not have inverses.<\/p>\n<p id=\"fs-id1165137655153\">Given a function[latex]\\,f\\left(x\\right),\\,[\/latex]we can verify whether some other function[latex]\\,g\\left(x\\right)\\,[\/latex]is the inverse of[latex]\\,f\\left(x\\right)\\,[\/latex]by checking whether either[latex]\\,g\\left(f\\left(x\\right)\\right)=x\\,[\/latex]or[latex]\\,f\\left(g\\left(x\\right)\\right)=x\\,[\/latex]is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)<\/p>\n<p id=\"fs-id1165135397975\">For example,[latex]\\,y=4x\\,[\/latex]and[latex]\\,y=\\frac{1}{4}x\\,[\/latex]are inverse functions.<\/p>\n<div id=\"fs-id1165137756798\" class=\"unnumbered aligncenter\">[latex]\\left({f}^{-1}\\circ f\\right)\\left(x\\right)={f}^{-1}\\left(4x\\right)=\\frac{1}{4}\\left(4x\\right)=x[\/latex]<\/div>\n<p id=\"fs-id1165137767233\">and<\/p>\n<div id=\"fs-id1165137755853\" class=\"unnumbered aligncenter\">[latex]\\left({f}^{}\\circ {f}^{-1}\\right)\\left(x\\right)=f\\left(\\frac{1}{4}x\\right)=4\\left(\\frac{1}{4}x\\right)=x[\/latex]<\/div>\n<p id=\"fs-id1165137438777\">A few coordinate pairs from the graph of the function[latex]\\,y=4x\\,[\/latex]are (\u22122, \u22128), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function[latex]\\,y=\\frac{1}{4}x\\,[\/latex]are (\u22128, \u22122), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.<\/p>\n<div id=\"fs-id1165137933105\" class=\"textbox key-takeaways\">\n<h3>Inverse Function<\/h3>\n<p id=\"fs-id1165137473076\">For any <span class=\"no-emphasis\">one-to-one function<\/span>[latex]\\,f\\left(x\\right)=y,\\,[\/latex]a function[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]is an inverse function of[latex]\\,f\\,[\/latex]if[latex]\\,{f}^{-1}\\left(y\\right)=x.\\,[\/latex]This can also be written as[latex]\\,{f}^{-1}\\left(f\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f.\\,[\/latex]It also follows that[latex]\\,f\\left({f}^{-1}\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,{f}^{-1}\\,[\/latex]if[latex]\\,{f}^{-1}\\,[\/latex]is the inverse of[latex]\\,f.\\,[\/latex]<\/p>\n<p id=\"fs-id1165137444821\">The notation [latex]{f}^{-1}[\/latex] is read [latex]\\text{\u201c}f[\/latex] inverse.\u201d Like any other function, we can use any variable name as the input for [latex]{f}^{-1},[\/latex] so we will often write[latex]\\,{f}^{-1}\\left(x\\right),[\/latex] which we read as [latex]\u201cf[\/latex] inverse of [latex]x.\u201d[\/latex]<br \/>\nKeep in mind that<\/p>\n<div id=\"fs-id1165137581324\" class=\"unnumbered aligncenter\">[latex]{f}^{-1}\\left(x\\right)\\ne \\frac{1}{f\\left(x\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165135194095\">and not all functions have inverses.<\/p>\n<\/div>\n<div id=\"Example_01_07_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137656641\">\n<div id=\"fs-id1165137922642\">\n<h3>Identifying an Inverse Function for a Given Input-Output Pair<\/h3>\n<p id=\"fs-id1165137659325\">If for a particular one-to-one function[latex]\\,f\\left(2\\right)=4\\,[\/latex]and[latex]\\,f\\left(5\\right)=12,\\,[\/latex]what are the corresponding input and output values for the inverse function?<\/p>\n<\/div>\n<div id=\"fs-id1165137768306\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137737081\">The inverse function reverses the input and output quantities, so if<\/p>\n<div id=\"fs-id1165137462459\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill f\\left(2\\right)& =& 4,\\text{ then }{f}^{-1}\\left(4\\right)=2;\\hfill \\\\ \\hfill f\\left(5\\right)& =& 12,{\\text{ then f}}^{-1}\\left(12\\right)=5.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137659464\">Alternatively, if we want to name the inverse function[latex]\\,g,\\,[\/latex]then[latex]\\,g\\left(4\\right)=2\\,[\/latex]and[latex]\\,g\\left(12\\right)=5.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165135245520\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135508518\">Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See <a class=\"autogenerated-content\" href=\"#Table_01_07_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_01_07_01\" summary=\"For (x,f(x)) we have the values (2, 4) and (5, 12); for (x, g(x)), we have the values (4, 2) and (12, 5).\">\n<thead>\n<tr>\n<th>[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\n<th>[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,12\\right)[\/latex]<\/td>\n<td>[latex]\\left(12,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635377\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_01\">\n<div id=\"fs-id1165137659088\">\n<p id=\"fs-id1165137659089\">Given that[latex]\\,{h}^{-1}\\left(6\\right)=2,\\,[\/latex]what are the corresponding input and output values of the original function[latex]\\,h?\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137645907\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137645908\">[latex]h\\left(2\\right)=6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134357354\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135434077\"><strong>Given two functions[latex]\\,\\,f\\left(x\\right)\\,\\,[\/latex]and[latex]\\,g\\left(x\\right),\\,[\/latex]test whether the functions are inverses of each other.<\/strong><\/p>\n<ol id=\"fs-id1165137452358\" type=\"1\">\n<li>Determine whether[latex]\\,f\\left(g\\left(x\\right)\\right)=x\\,[\/latex]or[latex]\\,g\\left(f\\left(x\\right)\\right)=x.[\/latex]<\/li>\n<li>If either statement is true, then both are true, and[latex]\\,g={f}^{-1}\\,[\/latex]and[latex]\\,f={g}^{-1}.\\,[\/latex]If either statement is false, then both are false, and[latex]\\,g\\ne {f}^{-1}\\,[\/latex]and[latex]\\,f\\ne {g}^{-1}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137557051\">\n<div id=\"fs-id1165137679032\">\n<h3>Testing Inverse Relationships Algebraically<\/h3>\n<p id=\"fs-id1165135519417\">If[latex]\\,f\\left(x\\right)=\\frac{1}{x+2}\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\frac{1}{x}-2,\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137627632\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165137675509\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill g\\left(f\\left(x\\right)\\right)& =& \\frac{1}{\\left(\\frac{1}{x+2}\\right)}-2\\hfill \\\\ & =& x+2-2\\hfill \\\\ & =& x\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137611481\">so<\/p>\n<div id=\"fs-id1165135678636\" class=\"unnumbered aligncenter\">[latex]g={f}^{-1}\\text{ and }f={g}^{-1}[\/latex]<\/div>\n<p id=\"fs-id1165135436648\">This is enough to answer yes to the question, but we can also verify the other formula.<\/p>\n<div id=\"fs-id1165137784350\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill f\\left(g\\left(x\\right)\\right)& =& \\frac{1}{\\frac{1}{x}-2+2}\\hfill \\\\ & =& \\frac{1}{\\frac{1}{x}}\\hfill \\\\ & =& x\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137733685\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135389000\">Notice the inverse operations are in reverse order of the operations from the original function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911663\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_02\">\n<div id=\"fs-id1165135160549\">\n<p id=\"fs-id1165135160550\">If[latex]\\,f\\left(x\\right)={x}^{3}-4\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\sqrt[\\,3]{x+4},\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137456449\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137600434\">Yes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135259560\">\n<div id=\"fs-id1165134042918\">\n<h3>Determining Inverse Relationships for Power Functions<\/h3>\n<p id=\"fs-id1165137441834\">If[latex]\\,f\\left(x\\right)={x}^{3}\\,[\/latex](the cube function) and[latex]\\,g\\left(x\\right)=\\frac{1}{3}x,\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137442603\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165137591632\" class=\"unnumbered aligncenter\">[latex]f\\left(g\\left(x\\right)\\right)=\\frac{{x}^{3}}{27}\\ne x[\/latex]<\/div>\n<p id=\"fs-id1165137694053\">No, the functions are not inverses.<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id1165135317479\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134192978\">The correct inverse to the cube is, of course, the cube root[latex]\\,\\sqrt[3]{x}={x}^{\\frac{1}{3}},\\,[\/latex]that is, the one-third is an exponent, not a multiplier.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806084\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_03\">\n<div id=\"fs-id1165135195489\">\n<p id=\"fs-id1165137573532\">If[latex]\\,f\\left(x\\right)={\\left(x-1\\right)}^{3}\\,\\text{and}\\,g\\left(x\\right)=\\sqrt[3]{x}+1,\\,[\/latex]is[latex]\\,g={f}^{-1}?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137501356\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137662080\">Yes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660004\" class=\"bc-section section\">\n<h3>Finding Domain and Range of Inverse Functions<\/h3>\n<p id=\"fs-id1165137591020\">The outputs of the function[latex]\\,f\\,[\/latex]are the inputs to[latex]\\,{f}^{-1},\\,[\/latex]so the range of[latex]\\,f\\,[\/latex]is also the domain of[latex]\\,{f}^{-1}.\\,[\/latex]Likewise, because the inputs to[latex]\\,f\\,[\/latex]are the outputs of[latex]\\,{f}^{-1},\\,[\/latex]the domain of[latex]\\,f\\,[\/latex]is the range of[latex]\\,{f}^{-1}.\\,[\/latex]We can visualize the situation as in <a class=\"autogenerated-content\" href=\"#Figure_01_07_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141943\/CNX_Precalc_Figure_01_07_003.jpg\" alt=\"Domain and range of a function and its inverse.\" width=\"487\" height=\"143\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3. <\/strong>Domain and range of a function and its inverse<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135557891\">When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of[latex]\\,f\\left(x\\right)=\\sqrt{x}\\,[\/latex]is[latex]\\,{f}^{-1}\\left(x\\right)={x}^{2},\\,[\/latex]because a square \u201cundoes\u201d a square root; but the square is only the inverse of the square root on the domain[latex]\\,\\left[0,\\infty \\right),\\,[\/latex]since that is the range of[latex]\\,f\\left(x\\right)=\\sqrt{x}.[\/latex]<\/p>\n<p id=\"fs-id1165137730185\">We can look at this problem from the other side, starting with the square (toolkit quadratic) function[latex]\\,f\\left(x\\right)={x}^{2}.\\,[\/latex]If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and \u20133. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the \u201cinverse\u201d is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.<\/p>\n<p id=\"fs-id1165137823552\">In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]with its domain limited to[latex]\\,\\left[0,\\infty \\right),\\,[\/latex]which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).<\/p>\n<p id=\"fs-id1165132037000\">If[latex]\\,f\\left(x\\right)={\\left(x-1\\right)}^{2}\\,[\/latex]on[latex]\\,\\left[1,\\infty \\right),\\,[\/latex]then the inverse function is[latex]\\,{f}^{-1}\\left(x\\right)=\\sqrt{x}+1.[\/latex]<\/p>\n<ul id=\"fs-id1165137851227\">\n<li>The domain of[latex]\\,f\\,[\/latex]= range of[latex]\\,{f}^{-1}\\,[\/latex]=[latex]\\,\\left[1,\\infty \\right).[\/latex]<\/li>\n<li>The domain of[latex]\\,{f}^{-1}\\,[\/latex]= range of[latex]\\,f\\,[\/latex]=[latex]\\,\\left[0,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<div id=\"fs-id1165137733804\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137723526\"><strong>Is it possible for a function to have more than one inverse?<\/strong><\/p>\n<p id=\"fs-id1165137456608\"><em>No. If two supposedly different functions, say,[latex]\\,g\\,[\/latex]and[latex]\\,h,\\,[\/latex]both meet the definition of being inverses of another function[latex]\\,f,\\,[\/latex]then you can prove that[latex]\\,g=h.\\,[\/latex]We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137704938\" class=\"textbox key-takeaways\">\n<h3>Domain and Range of Inverse Functions<\/h3>\n<p id=\"fs-id1165135319550\">The range of a function[latex]\\,f\\left(x\\right)\\,[\/latex]is the domain of the inverse function[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137673886\">The domain of[latex]\\,f\\left(x\\right)\\,[\/latex]is the range of[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135308785\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137605040\"><strong>Given a function, find the domain and range of its inverse.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165137530434\" type=\"1\">\n<li>If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse.<\/li>\n<li>If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_05\" class=\"textbox examples\">\n<h3>Finding the Inverses of Toolkit Functions<\/h3>\n<p id=\"fs-id1165137448020\">Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in <a class=\"autogenerated-content\" href=\"#Table_01_07_02\">(Figure)<\/a>. We restrict the domain in such a fashion that the function assumes all <em>y<\/em>-values exactly once.<\/p>\n<table id=\"Table_01_07_02\" summary=\"A list of the toolkit function. The constant function is f(x) = c where c is the constant; the identity function is f(x) = x; the absolute function is f(x)=|x|; the quadratic function is f(x) = x^2; the cubic function is f(x)=x^3; the reciprocal function is f(x)=1\/x; the reciprocal squared function is f(x)=1\/x^2; the square root function is f(x)=sqrt(x); the cube root function is f(x) = x^(1\/3).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th>Constant<\/th>\n<th>Identity<\/th>\n<th>Quadratic<\/th>\n<th>Cubic<\/th>\n<th>Reciprocal<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f\\left(x\\right)=c[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Reciprocal squared<\/strong><\/td>\n<td><strong>Cube root<\/strong><\/td>\n<td><strong>Square root<\/strong><\/td>\n<td><strong>Absolute value<\/strong><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137767030\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132988445\">The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no inverse.<\/p>\n<p>The absolute value function can be restricted to the domain[latex]\\,\\left[0,\\infty \\right),[\/latex]where it is equal to the identity function.<\/p>\n<p id=\"fs-id1165137642849\">The reciprocal-squared function can be restricted to the domain[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id1165137901280\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137742302\">We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_004\">(Figure)<\/a>. They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes the horizontal line test, then in that restricted domain, it can have an inverse.<\/p>\n<div id=\"Figure_01_07_004\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141957\/CNX_Precalc_Figure_01_07_004ab.jpg\" alt=\"Graph of an absolute function.\" width=\"975\" height=\"404\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong> (a) Absolute value (b) Reciprocal square<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137544599\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_09\">\n<div id=\"fs-id1165135406939\">\n<p id=\"fs-id1165137507853\">The domain of function[latex]\\,f\\,[\/latex]is[latex]\\,\\left(1,\\infty \\right)\\,[\/latex]and the range of function[latex]\\,f\\,[\/latex]is[latex]\\,\\left(\\mathrm{-\\infty },-2\\right).\\,[\/latex]Find the domain and range of the inverse function.<\/p>\n<\/div>\n<div id=\"fs-id1165137431277\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137451713\">The domain of function[latex]\\,{f}^{-1}\\,[\/latex]is[latex]\\,\\left(-\\infty \\text{,}-2\\right)\\,[\/latex]and the range of function[latex]\\,{f}^{-1}\\,[\/latex]is[latex]\\,\\left(1,\\infty \\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619159\" class=\"bc-section section\">\n<h3>Finding and Evaluating Inverse Functions<\/h3>\n<p id=\"fs-id1165137761017\">Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.<\/p>\n<div id=\"fs-id1165135466392\" class=\"bc-section section\">\n<h4>Inverting Tabular Functions<\/h4>\n<p id=\"fs-id1165135190714\">Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.<\/p>\n<p id=\"fs-id1165137422578\">Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.<\/p>\n<div id=\"Example_01_07_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135544995\">\n<div id=\"fs-id1165137698262\">\n<h3>Interpreting the Inverse of a Tabular Function<\/h3>\n<p id=\"fs-id1165135435474\">A function[latex]\\,f\\left(t\\right)\\,[\/latex]is given in <a class=\"autogenerated-content\" href=\"#Table_01_07_03\">(Figure)<\/a>, showing distance in miles that a car has traveled in[latex]\\,t\\,[\/latex]minutes. Find and interpret[latex]\\,{f}^{-1}\\left(70\\right).[\/latex]<\/p>\n<table id=\"Table_01_07_03\" summary=\"Two rows and five columns. The first row is labeled \u201ct (minutes)\u201d, and the second row is labeled \u201cf(x) (miles)\u201d. Reading the columns as ordered pairs, we have the following values (30, 20), (50, 40), (70, 60), and (90, 70).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]t\\text{ (minutes)}[\/latex]<\/strong><\/td>\n<td>30<\/td>\n<td>50<\/td>\n<td>70<\/td>\n<td>90<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex]<\/strong><\/td>\n<td>20<\/td>\n<td>40<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137407569\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137640334\">The inverse function takes an output of[latex]\\,f\\,[\/latex]and returns an input for[latex]\\,f.\\,[\/latex]So in the expression[latex]\\,{f}^{-1}\\left(70\\right),\\,[\/latex]70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function[latex]\\,f,\\,[\/latex]90 minutes, so[latex]\\,{f}^{-1}\\left(70\\right)=90.\\,[\/latex]The interpretation of this is that, to drive 70 miles, it took 90 minutes.<\/p>\n<p id=\"fs-id1165135181841\">Alternatively, recall that the definition of the inverse was that if[latex]\\,f\\left(a\\right)=b,\\,[\/latex]then[latex]\\,{f}^{-1}\\left(b\\right)=a.\\,[\/latex]By this definition, if we are given[latex]\\,{f}^{-1}\\left(70\\right)=a,\\,[\/latex]then we are looking for a value[latex]\\,a\\,[\/latex]so that[latex]\\,f\\left(a\\right)=70.\\,[\/latex]In this case, we are looking for a[latex]\\,t\\,[\/latex]so that[latex]\\,f\\left(t\\right)=70,\\,[\/latex]which is when[latex]\\,t=90.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135169494\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_04\">\n<div id=\"fs-id1165135443767\">\n<p id=\"fs-id1165134108483\">Using <a class=\"autogenerated-content\" href=\"#Table_01_07_04\">(Figure)<\/a>, find and interpret (a)[latex]\\text{ }f\\left(60\\right),[\/latex]and (b)[latex]\\text{ }{f}^{-1}\\left(60\\right).[\/latex]<\/p>\n<table id=\"Table_01_07_04\" summary=\"Two rows and five columns. The first row is labeled \u201ct (minutes)\u201d, and the second row is labeled \u201cf(t)\u201d. Reading the columns as ordered pairs, we have the following values (30, 20), (50, 40), (70, 60), and (90, 70).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]t\\text{ (minutes)}[\/latex]<\/strong><\/td>\n<td>30<\/td>\n<td>50<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<td>90<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex]<\/strong><\/td>\n<td>20<\/td>\n<td>40<\/td>\n<td>50<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137862841\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165137862843\" type=\"a\">\n<li>[latex]f\\left(60\\right)=50.\\,[\/latex]In 60 minutes, 50 miles are traveled.<\/li>\n<li>[latex]{f}^{-1}\\left(60\\right)=70.\\,[\/latex]To travel 60 miles, it will take 70 minutes.<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137418615\" class=\"bc-section section\">\n<h4>Evaluating the Inverse of a Function, Given a Graph of the Original Function<\/h4>\n<p id=\"fs-id1165137400045\">We saw in <a class=\"target-chapter\" href=\"\/contents\/55f2e8ec-a982-4586-9d48-a2f43d7b4107\">Functions and Function Notation<\/a> that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the <em>vertical<\/em> extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the <em>horizontal<\/em> extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function\u2019s graph.<\/p>\n<div id=\"fs-id1165133045388\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135333128\"><strong>Given the graph of a function, evaluate its inverse at specific points.<\/strong><\/p>\n<ol id=\"fs-id1165137464840\" type=\"1\">\n<li>Find the desired input on the <em>y<\/em>-axis of the given graph.<\/li>\n<li>Read the inverse function\u2019s output from the <em>x<\/em>-axis of the given graph.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_07\" class=\"textbox examples\">\n<div id=\"fs-id1165135434803\">\n<div id=\"fs-id1165135434805\">\n<h3>Evaluating a Function and Its Inverse from a Graph at Specific Points<\/h3>\n<p id=\"fs-id1165134108622\">A function[latex]\\,g\\left(x\\right)\\,[\/latex]is given in <a class=\"autogenerated-content\" href=\"#Figure_01_07_006\">(Figure)<\/a>. Find[latex]\\,g\\left(3\\right)\\,[\/latex]and[latex]\\,{g}^{-1}\\left(3\\right).[\/latex]<\/p>\n<div id=\"Figure_01_07_006\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142008\/CNX_Precalc_Figure_01_07_006.jpg\" alt=\"Graph of g(x).\" width=\"487\" height=\"254\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137468840\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137468842\">To evaluate [latex]g\\left(3\\right),\\,[\/latex]we find 3 on the <em>x<\/em>-axis and find the corresponding output value on the <em>y<\/em>-axis. The point [latex]\\,\\left(3,1\\right)\\,[\/latex]tells us that[latex]\\,g\\left(3\\right)=1.[\/latex]<\/p>\n<p id=\"fs-id1165137405078\">To evaluate[latex]\\,{g}^{-1}\\left(3\\right),\\,[\/latex]recall that by definition[latex]\\,{g}^{-1}\\left(3\\right)\\,[\/latex]means the value of <em>x<\/em> for which[latex]\\,g\\left(x\\right)=3.\\,[\/latex]By looking for the output value 3 on the vertical axis, we find the point[latex]\\,\\left(5,3\\right)\\,[\/latex]on the graph, which means[latex]\\,g\\left(5\\right)=3,\\,[\/latex]so by definition,[latex]\\,{g}^{-1}\\left(3\\right)=5.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_01_07_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_007\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142010\/CNX_Precalc_Figure_01_07_007.jpg\" alt=\"Graph of g(x).\" width=\"487\" height=\"254\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137667918\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_05\">\n<div id=\"fs-id1165137812559\">\n<p id=\"fs-id1165137812560\">Using the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_07_007\">(Figure)<\/a>, (a) find[latex]\\,{g}^{-1}\\left(1\\right),[\/latex]and (b) estimate[latex]\\,{g}^{-1}\\left(4\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135528890\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135528891\">a. 3; b. 5.6<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605437\" class=\"bc-section section\">\n<h4>Finding Inverses of Functions Represented by Formulas<\/h4>\n<p id=\"fs-id1165137433184\">Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula\u2014for example,[latex]\\,y\\,[\/latex]as a function of[latex]\\,x\\text{\u2014}[\/latex]we can often find the inverse function by solving to obtain[latex]\\,x\\,[\/latex]as a function of[latex]\\,y.[\/latex]<\/p>\n<div id=\"fs-id1165137652548\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135195849\"><strong>Given a function represented by a formula, find the inverse.<\/strong><\/p>\n<ol id=\"fs-id1165135443898\" type=\"1\">\n<li>Make sure[latex]\\,f\\,[\/latex]is a one-to-one function.<\/li>\n<li>Solve for[latex]\\,x.[\/latex]<\/li>\n<li>Interchange[latex]\\,x\\,[\/latex]and[latex]\\,y.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_07_08\" class=\"textbox examples\">\n<div id=\"fs-id1165135186316\">\n<div id=\"fs-id1165135186318\">\n<h3>Inverting the Fahrenheit-to-Celsius Function<\/h3>\n<p id=\"fs-id1165137596585\">Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.<\/p>\n<div id=\"fs-id1165133306998\" class=\"unnumbered aligncenter\">[latex]C=\\frac{5}{9}\\left(F-32\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135417800\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165135193737\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill C& =& \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\ \\hfill C\\cdot \\frac{9}{5}& =& F-32\\hfill \\\\ \\hfill F& =& \\frac{9}{5}C+32\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137819987\">By solving in general, we have uncovered the inverse function. If<\/p>\n<div id=\"fs-id1165135173380\" class=\"unnumbered aligncenter\">[latex]C=h\\left(F\\right)=\\frac{5}{9}\\left(F-32\\right),[\/latex]<\/div>\n<p id=\"fs-id1165135435603\">then<\/p>\n<div id=\"fs-id1165137772327\" class=\"unnumbered aligncenter\">[latex]F={h}^{-1}\\left(C\\right)=\\frac{9}{5}C+32[\/latex]<\/div>\n<p id=\"fs-id1165137573279\">In this case, we introduced a function[latex]\\,h\\,[\/latex]to represent the conversion because the input and output variables are descriptive, and writing[latex]\\,{C}^{-1}\\,[\/latex]could get confusing.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_06\">\n<div id=\"fs-id1165135563330\">\n<p id=\"fs-id1165135563331\">Solve for[latex]\\,x\\,[\/latex]in terms of[latex]\\,y\\,[\/latex]given[latex]\\,y=\\frac{1}{3}\\left(x-5\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134049417\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134049418\">[latex]x=3y+5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134065146\">\n<div id=\"fs-id1165137409366\">\n<h3>Solving to Find an Inverse Function<\/h3>\n<p id=\"fs-id1165137891504\">Find the inverse of the function[latex]\\,f\\left(x\\right)=\\frac{2}{x-3}+4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137482074\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165135189953\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill y& =& \\frac{2}{x-3}+4\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Set up an equation}.\\hfill \\\\ \\hfill y-4& =& \\frac{2}{x-3}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Subtract 4 from both sides}.\\hfill \\\\ \\hfill x-3& =& \\frac{2}{y-4}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply both sides by }x-3\\text{ and divide by }y-4.\\hfill \\\\ \\hfill x& =& \\frac{2}{y-4}+3\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Add 3 to both sides}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137678168\">So[latex]\\,{f}^{-1}\\left(y\\right)=\\frac{2}{y-4}+3\\,[\/latex]or[latex]\\,{f}^{-1}\\left(x\\right)=\\frac{2}{x-4}+3.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137864156\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135394231\">The domain and range of[latex]\\,f\\,[\/latex]exclude the values 3 and 4, respectively.[latex]\\,f\\,[\/latex] and [latex]\\,{f}^{-1}\\,[\/latex]are equal at two points but are not the same function, as we can see by creating <a class=\"autogenerated-content\" href=\"#Table_01_07_05\">(Figure)<\/a>.<\/p>\n<table id=\"Table_01_07_05\" summary=\"The values of f(x) are: f(1)=3, f(2)=2, and f(5)=5. So f^(-1)(y)=y.\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<td>[latex]{f}^{-1}\\left(y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>3<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<td>[latex]y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137603677\">\n<div id=\"fs-id1165137547656\">\n<h3>Solving to Find an Inverse with Radicals<\/h3>\n<p>Find the inverse of the function[latex]\\,f\\left(x\\right)=2+\\sqrt{x-4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135193684\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165137828173\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill y& =& 2+\\sqrt{x-4}\\hfill \\\\ \\hfill {\\left(y-2\\right)}^{2}& =& x-4\\hfill \\\\ \\hfill x& =& {\\left(y-2\\right)}^{2}+4\\hfill \\end{array}[\/latex]<\/div>\n<p>So[latex]\\,{f}^{-1}\\left(x\\right)={\\left(x-2\\right)}^{2}+4.[\/latex]<\/p>\n<p id=\"fs-id1165137900392\">The domain of[latex]\\,f\\,[\/latex]is[latex]\\,\\left[4,\\infty \\right).\\,[\/latex]Notice that the range of[latex]\\,f\\,[\/latex]is[latex]\\,\\left[2,\\infty \\right),\\,[\/latex]so this means that the domain of the inverse function[latex]\\,{f}^{-1}\\,[\/latex]is also[latex]\\,\\left[2,\\infty \\right).[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137667328\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135546050\">The formula we found for[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]looks like it would be valid for all real[latex]\\,x.\\,[\/latex]However,[latex]\\,{f}^{-1}\\,[\/latex]itself must have an inverse (namely,[latex]\\,f\\,[\/latex]) so we have to restrict the domain of[latex]\\,{f}^{-1}\\,[\/latex]to[latex]\\,\\left[2,\\infty \\right)\\,[\/latex]in order to make[latex]\\,{f}^{-1}\\,[\/latex]a one-to-one function. This domain of[latex]\\,{f}^{-1}\\,[\/latex]is exactly the range of[latex]\\,f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640068\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_07\">\n<div id=\"fs-id1165137756073\">\n<p id=\"fs-id1165137756074\">What is the inverse of the function[latex]\\,f\\left(x\\right)=2-\\sqrt{x}?[\/latex]State the domains of both the function and the inverse function.<\/p>\n<\/div>\n<div id=\"fs-id1165137937549\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137937550\">[latex]{f}^{-1}\\left(x\\right)={\\left(2-x\\right)}^{2};\\,\\,\\text{domain}\\,\\,\\text{of}\\,\\,f:\\,\\,\\left[0,\\infty \\right);\\,\\,\\text{domain}\\,\\,\\text{of}\\,\\,{f}^{-1}:\\,\\,\\left(-\\infty ,2\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473011\" class=\"bc-section section\">\n<h3>Finding Inverse Functions and Their Graphs<\/h3>\n<p id=\"fs-id1165137463843\">Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]restricted to the domain[latex]\\,\\left[0,\\infty \\right)\\text{,}[\/latex] on which this function is one-to-one, and graph it as in <a class=\"autogenerated-content\" href=\"#Figure_01_07_008\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_008\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142016\/CNX_Precalc_Figure_01_07_008.jpg\" alt=\"Graph of f(x).\" width=\"487\" height=\"254\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7. <\/strong>Quadratic function with domain restricted to [0, \u221e).<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137419977\"><span class=\"no-emphasis\">Restricting the domain<\/span> to[latex]\\,\\left[0,\\infty \\right)\\,[\/latex]makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.<\/p>\n<p id=\"fs-id1165137656093\">We already know that the inverse of the toolkit quadratic function is the square root function, that is, [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}.[\/latex] What happens if we graph both [latex]f\\text{ }[\/latex] and [latex]{f}^{-1}[\/latex] on the same set of axes, using the [latex]x\\text{-}[\/latex]axis for the input to both [latex]f\\text{ and }{f}^{-1}?[\/latex]<\/p>\n<p id=\"fs-id1165131968090\">We notice a distinct relationship: The graph of[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]is the graph of[latex]\\,f\\left(x\\right)\\,[\/latex]reflected about the diagonal line[latex]\\,y=x,\\,[\/latex]which we will call the identity line, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_009\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_009\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142034\/CNX_Precalc_Figure_01_07_009.jpg\" alt=\"Graph of f(x) and f^(-1)(x).\" width=\"487\" height=\"251\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8. <\/strong>Square and square-root functions on the non-negative domain<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137393212\">This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.<\/p>\n<div id=\"Example_01_07_11\" class=\"textbox examples\">\n<div id=\"fs-id1165134430460\">\n<div id=\"fs-id1165134430463\">\n<h3>Finding the Inverse of a Function Using Reflection about the Identity Line<\/h3>\n<p id=\"fs-id1165134342627\">Given the graph of[latex]\\,f\\left(x\\right)\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Figure_01_07_010\">(Figure)<\/a>, sketch a graph of[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<div id=\"Figure_01_07_010\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142046\/CNX_Precalc_Figure_01_07_010.jpg\" alt=\"Graph of f^(-1)(x).\" width=\"487\" height=\"363\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407658\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137407660\">This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of[latex]\\,\\left(0,\\infty \\right)\\,[\/latex]and range of[latex]\\,\\left(-\\infty ,\\infty \\right),\\,[\/latex]so the inverse will have a domain of[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]and range of[latex]\\,\\left(0,\\infty \\right).[\/latex]<\/p>\n<p>If we reflect this graph over the line[latex]\\,y=x,\\,[\/latex]the point[latex]\\,\\left(1,0\\right)\\,[\/latex]reflects to[latex]\\,\\left(0,1\\right)\\,[\/latex]and the point[latex]\\,\\left(4,2\\right)\\,[\/latex]reflects to[latex]\\,\\left(2,4\\right).\\,[\/latex]Sketching the inverse on the same axes as the original graph gives <a class=\"autogenerated-content\" href=\"#Figure_01_07_011\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142103\/CNX_Precalc_Figure_01_07_011.jpg\" alt=\"Graph of f(x) and f^(-1)(x).\" width=\"487\" height=\"363\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10. <\/strong>The function and its inverse, showing reflection about the identity line<\/figcaption><\/figure>\n<p id=\"fs-id1165137416305\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187125\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_07_08\">\n<div id=\"fs-id1165137619929\">\n<p id=\"fs-id1165137619930\">Draw graphs of the functions[latex]\\,f\\text{ }[\/latex]and[latex]\\text{ }{f}^{-1}[\/latex]from <a class=\"autogenerated-content\" href=\"#Example_01_07_09\">(Figure)<\/a>.<\/p>\n<\/div>\n<div id=\"fs-id1165137911739\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137539140\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142110\/CNX_Precalc_Figure_01_07_012.jpg\" alt=\"Graph of f(x) and f^(-1)(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627081\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165134388228\"><strong>Is there any function that is equal to its own inverse?<\/strong><\/p>\n<p id=\"fs-id1165137602656\"><em>Yes. If[latex]\\,f={f}^{-1},\\,[\/latex]then[latex]\\,f\\left(f\\left(x\\right)\\right)=x,\\,[\/latex]and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because<\/em><\/p>\n<div id=\"fs-id1165135205827\" class=\"unnumbered aligncenter\">[latex]\\frac{1}{\\frac{1}{x}}=x[\/latex]<\/div>\n<p id=\"fs-id1165137897050\"><em>Any function[latex]\\,f\\left(x\\right)=c-x,\\,[\/latex]where[latex]\\,c\\,[\/latex]is a constant, is also equal to its own inverse.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137410410\" class=\"precalculus media\">\n<p id=\"fs-id1165137410501\">Access these online resources for additional instruction and practice with inverse functions.<\/p>\n<ul id=\"fs-id1165137582034\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/inversefunction\">Inverse Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/onetoone\">One-to-one Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/inversfuncgraph\">Inverse Function Values Using Graph<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/restrictdomain\">Restricting the Domain and Finding the Inverse<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-10\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC01\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n<\/div>\n<div id=\"fs-id1165137591826\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135536334\">\n<li>If[latex]\\,g\\left(x\\right)\\,[\/latex]is the inverse of[latex]\\,f\\left(x\\right),\\,[\/latex]then[latex]\\,g\\left(f\\left(x\\right)\\right)=f\\left(g\\left(x\\right)\\right)=x.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_01_07_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_01_07_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_01_07_03\">(Figure)<\/a>.<\/li>\n<li>Only some of the toolkit functions have an inverse. See <a class=\"autogenerated-content\" href=\"#Example_01_07_05\">(Figure)<\/a>.<\/li>\n<li>For a function to have an inverse, it must be one-to-one (pass the horizontal line test).<\/li>\n<li>A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.<\/li>\n<li>For a tabular function, exchange the input and output rows to obtain the inverse. See <a class=\"autogenerated-content\" href=\"#Example_01_07_06\">(Figure)<\/a>.<\/li>\n<li>The inverse of a function can be determined at specific points on its graph. See <a class=\"autogenerated-content\" href=\"#Example_01_07_07\">(Figure)<\/a>.<\/li>\n<li>To find the inverse of a formula, solve the equation[latex]\\,y=f\\left(x\\right)\\,[\/latex]for[latex]\\,x\\,[\/latex]as a function of[latex]\\,y.\\,[\/latex]Then exchange the labels[latex]\\,x\\,[\/latex]and[latex]\\,\\,y.\\,\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_01_07_08\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_01_07_09\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_01_07_10\">(Figure)<\/a>.<\/li>\n<li>The graph of an inverse function is the reflection of the graph of the original function across the line[latex]\\,y=x.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_01_07_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137871042\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135187563\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137407341\">\n<div id=\"fs-id1165135193086\">\n<p id=\"fs-id1165135193088\">Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?<\/p>\n<\/div>\n<div id=\"fs-id1165137517264\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134061896\">Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that[latex]\\,y[\/latex]-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no[latex]\\,y[\/latex]-values repeat and the function is one-to-one.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137408636\">\n<div id=\"fs-id1165137408638\">\n<p id=\"fs-id1165134113962\">Why do we restrict the domain of the function[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]to find the function\u2019s inverse?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137389621\">\n<div id=\"fs-id1165135400199\">\n<p id=\"fs-id1165135400201\">Can a function be its own inverse? Explain.<\/p>\n<\/div>\n<div id=\"fs-id1165137553896\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137654653\">Yes. For example,[latex]\\,f\\left(x\\right)=\\frac{1}{x}\\,[\/latex]is its own inverse.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188794\">\n<div id=\"fs-id1165137564806\">\n<p id=\"fs-id1165137564808\">Are one-to-one functions either always increasing or always decreasing? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419050\">\n<div id=\"fs-id1165137932403\">\n<p id=\"fs-id1165137932405\">How do you find the inverse of a function algebraically?<\/p>\n<\/div>\n<div id=\"fs-id1165137785042\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137673500\">Given a function[latex]\\,y=f\\left(x\\right),\\,[\/latex]solve for[latex]\\,x\\,[\/latex]in terms of[latex]\\,y.\\,[\/latex]Interchange the[latex]\\,x\\,[\/latex]and[latex]\\,y.\\,[\/latex]Solve the new equation for[latex]\\,y.\\,[\/latex]The expression for[latex]\\,y\\,[\/latex]is the inverse,[latex]\\,y={f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836714\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165137422830\">\n<div id=\"fs-id1165137806758\">\n<p id=\"fs-id1165137806761\">Show that the function[latex]\\,f\\left(x\\right)=a-x\\,[\/latex]is its own inverse for all real numbers[latex]\\,a.\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137469451\">For the following exercises, find[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]for each function.<\/p>\n<div id=\"fs-id1165134312158\">\n<div id=\"fs-id1165134312161\">\n<p id=\"fs-id1165137562307\">[latex]f\\left(x\\right)=x+3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135196794\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135196796\">[latex]{f}^{-1}\\left(x\\right)=x-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634366\">\n<div id=\"fs-id1165137634368\">\n<p id=\"fs-id1165137679711\">[latex]f\\left(x\\right)=x+5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706135\">\n<div id=\"fs-id1165137422592\">\n<p id=\"fs-id1165137422594\">[latex]f\\left(x\\right)=2-x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137812372\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137812374\">[latex]{f}^{-1}\\left(x\\right)=2-x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137561652\">\n<div id=\"fs-id1165137653456\">\n<p id=\"fs-id1165137653458\">[latex]f\\left(x\\right)=3-x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600416\">\n<div id=\"fs-id1165135198605\">\n<p id=\"fs-id1165135198608\">[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135541959\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135541961\">[latex]{f}^{-1}\\left(x\\right)=\\frac{-2x}{x-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137541591\">\n<div>\n<p id=\"fs-id1165134043733\">[latex]f\\left(x\\right)=\\frac{2x+3}{5x+4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137410615\">For the following exercises, find a domain on which each function[latex]\\,f\\,[\/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\\,f\\,[\/latex]restricted to that domain.<\/p>\n<div id=\"fs-id1165134148519\">\n<div id=\"fs-id1165134148521\">\n<p id=\"fs-id1165137737103\">[latex]f\\left(x\\right)={\\left(x+7\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137849508\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137655378\">domain of[latex]f\\left(x\\right):\\,\\left[-7,\\infty \\right);\\,{f}^{-1}\\left(x\\right)=\\sqrt{x}-7[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862703\">\n<div id=\"fs-id1165137531119\">\n<p id=\"fs-id1165137531121\">[latex]f\\left(x\\right)={\\left(x-6\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137938700\">\n<p id=\"fs-id1165134042196\">[latex]f\\left(x\\right)={x}^{2}-5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137603366\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137603368\">domain of[latex]\\,f\\left(x\\right):\\,\\left[0,\\infty \\right);\\,{f}^{-1}\\left(x\\right)=\\sqrt{x+5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734966\">\n<div id=\"fs-id1165137734969\">\n<p id=\"fs-id1165137731914\">Given[latex]\\,f\\left(x\\right)=\\frac{x}{2+x}\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\frac{2x}{1-x}:[\/latex]<\/p>\n<ol id=\"fs-id1165137838751\" type=\"a\">\n<li>Find[latex]\\,f\\left(g\\left(x\\right)\\right)\\,[\/latex]and[latex]\\,g\\left(f\\left(x\\right)\\right).[\/latex]<\/li>\n<li>What does the answer tell us about the relationship between[latex]\\,f\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137730082\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137730084\">a.[latex]\\,f\\left(g\\left(x\\right)\\right)=x\\,[\/latex]and[latex]\\,g\\left(f\\left(x\\right)\\right)=x.\\,[\/latex]b. This tells us that[latex]\\,f\\,[\/latex]and[latex]\\,g\\,[\/latex]are inverse functions<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137444427\">For the following exercises, use function composition to verify that[latex]\\,f\\left(x\\right)\\,[\/latex]and[latex]\\,g\\left(x\\right)\\,[\/latex]are inverse functions.<\/p>\n<div id=\"fs-id1165137437578\">\n<div id=\"fs-id1165137619341\">\n<p id=\"fs-id1165137619343\">[latex]f\\left(x\\right)=\\sqrt[3]{x-1}\\,[\/latex]and[latex]\\,g\\left(x\\right)={x}^{3}+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137883792\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135310692\">[latex]f\\left(g\\left(x\\right)\\right)=x,\\,g\\left(f\\left(x\\right)\\right)=x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538749\">\n<div id=\"fs-id1165135538751\">\n<p id=\"fs-id1165137452674\">[latex]f\\left(x\\right)=-3x+5\\,[\/latex]and[latex]\\,g\\left(x\\right)=\\frac{x-5}{-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188614\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135176520\">For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n<div id=\"fs-id1165137645254\">\n<div>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/div>\n<div id=\"fs-id1165134113903\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137534912\">one-to-one<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173420\">\n<div id=\"fs-id1165135173423\">\n<p id=\"fs-id1165135256110\">[latex]f\\left(x\\right)=\\sqrt[3]{3x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135424683\">\n<div id=\"fs-id1165137408415\">\n<p id=\"fs-id1165137408417\">[latex]f\\left(x\\right)=-5x+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137433244\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137393282\">one-to-one<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704606\">\n<div id=\"fs-id1165137704608\">\n<p id=\"fs-id1165137837047\">[latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137528556\">For the following exercises, determine whether the graph represents a one-to-one function.<\/p>\n<div id=\"fs-id1165137528559\">\n<div id=\"fs-id1165135251340\"><span id=\"fs-id1165135341390\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142115\/CNX_Precalc_Figure_01_07_201.jpg\" alt=\"Graph of a parabola.\" \/><\/span><\/div>\n<div id=\"fs-id1165137612243\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131959461\">not one-to-one<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192971\">\n<div id=\"fs-id1165135192973\"><span id=\"fs-id1165134379457\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142130\/CNX_Precalc_Figure_01_07_202.jpg\" alt=\"Graph of a step-function.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165137849556\">For the following exercises, use the graph of[latex]\\,f\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_203\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_203\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137855139\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142133\/CNX_Precalc_Figure_01_07_203.jpg\" alt=\"Graph of a line.\" \/><\/span><\/div>\n<div id=\"fs-id1165137863913\">\n<div id=\"fs-id1165137854842\">\n<p id=\"fs-id1165137854844\">Find[latex]\\,f\\left(0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137653684\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137851374\">[latex]3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417814\">\n<div id=\"fs-id1165137417816\">\n<p id=\"fs-id1165137417818\">Solve[latex]\\,f\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133093360\">\n<div id=\"fs-id1165133093363\">\n<p id=\"fs-id1165137461116\">Find[latex]\\,{f}^{-1}\\left(0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137573270\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137573273\">[latex]2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451393\">\n<div id=\"fs-id1165137806036\">\n<p id=\"fs-id1165137806038\">Solve[latex]\\,{f}^{-1}\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135639868\">For the following exercises, use the graph of the one-to-one function shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_204\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_204\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137611817\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142135\/CNX_Precalc_Figure_01_07_204.jpg\" alt=\"Graph of a square root function.\" \/><\/span><\/div>\n<div id=\"fs-id1165137884386\">\n<div id=\"fs-id1165137812123\">\n<p id=\"fs-id1165137812125\">Sketch the graph of[latex]\\,{f}^{-1}.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134113897\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137939427\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142137\/CNX_Precalc_Figure_01_07_205.jpg\" alt=\"Graph of a square root function and its inverse.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137892244\">\n<div id=\"fs-id1165137603262\">\n<p id=\"fs-id1165137603264\">Find[latex]\\,f\\left(6\\right)\\text{ and }{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137827763\">\n<div id=\"fs-id1165137827765\">\n<p id=\"fs-id1165137827767\">If the complete graph of[latex]\\,f\\,[\/latex]is shown, find the domain of[latex]\\,f.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134259264\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135699154\">[latex]\\left[2,10\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137644584\">\n<p id=\"fs-id1165135307895\">If the complete graph of[latex]\\,f\\,[\/latex]is shown, find the range of[latex]\\,f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135434734\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137781583\">For the following exercises, evaluate or solve, assuming that the function[latex]\\,f\\,[\/latex]is one-to-one.<\/p>\n<div id=\"fs-id1165137771148\">\n<div id=\"fs-id1165135250622\">\n<p id=\"fs-id1165135250624\">If[latex]\\,f\\left(6\\right)=7,\\,[\/latex]find[latex]\\,\\,{f}^{-1}\\left(7\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137714192\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137714194\">[latex]6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805757\">\n<div id=\"fs-id1165135190808\">\n<p id=\"fs-id1165135190810\">If[latex]\\,f\\left(3\\right)=2,\\,[\/latex]find[latex]\\,{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640685\">\n<div id=\"fs-id1165137640687\">\n<p id=\"fs-id1165137640689\">If[latex]\\,{f}^{-1}\\left(-4\\right)=-8,\\,[\/latex]find[latex]\\,f\\left(-8\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137848946\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135169413\">[latex]-4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404972\">\n<div id=\"fs-id1165137404974\">\n<p id=\"fs-id1165137404976\">If[latex]\\,{f}^{-1}\\left(-2\\right)=-1,\\,[\/latex]find[latex]\\,f\\left(-1\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135195398\">For the following exercises, use the values listed in <a class=\"autogenerated-content\" href=\"#Table_01_07_06\">(Figure)<\/a> to evaluate or solve.<\/p>\n<table id=\"Table_01_07_06\" summary=\"Two columns and ten rows. The first column is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So for f(0)=8, f(1)=0, f(2)=7, f(3)=4, f(4)=2, f(5)=6, f(6)=5, f(7)=8, f(8)=9, and f(9)=1.\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137400581\">\n<div id=\"fs-id1165137400584\">\n<p id=\"fs-id1165137400586\">Find[latex]\\,f\\left(1\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135263655\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137823491\">[latex]0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137807065\">\n<div id=\"fs-id1165137807068\">\n<p id=\"fs-id1165137645884\">Solve[latex]\\,f\\left(x\\right)=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736914\">\n<div id=\"fs-id1165137736916\">\n<p id=\"fs-id1165137871008\">Find[latex]\\,{f}^{-1}\\left(0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137422471\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137422474\">[latex]\\,1\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137686729\">\n<div id=\"fs-id1165137686731\">\n<p id=\"fs-id1165137686733\">Solve[latex]\\,{f}^{-1}\\left(x\\right)=7.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135209686\">\n<p id=\"fs-id1165135209688\">Use the tabular representation of[latex]\\,f\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Table_01_07_08\">(Figure)<\/a> to create a table for[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<table id=\"Table_01_07_08\" summary=\"Two rows and six columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 3, 6, 9, 13, and 14. So for f(3)=1, f(6)=4, f(9)=7, f(13)=12, and f(14)=16.\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>3<\/td>\n<td>6<\/td>\n<td>9<\/td>\n<td>13<\/td>\n<td>14<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>1<\/td>\n<td>4<\/td>\n<td>7<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137406963\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id1165134138595\" class=\"unnumbered\" summary=\"Two rows and six columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf^(-1)(x)\u201d. The values of x are 1, 4, 7, 12, and 16. So for f^(-1) (1)=1, f^(-1) (4)=6, f^(-1) (7)=9, f^(-1) (12)=13, and f^(-1)f(16)=14.\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>1<\/td>\n<td>4<\/td>\n<td>7<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>[latex]{f}^{-1}\\left(x\\right)[\/latex]<\/td>\n<td>3<\/td>\n<td>6<\/td>\n<td>9<\/td>\n<td>13<\/td>\n<td>14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641552\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137401884\">For the following exercises, find the inverse function. Then, graph the function and its inverse.<\/p>\n<div id=\"fs-id1165137401888\">\n<div id=\"fs-id1165137470994\">\n<p id=\"fs-id1165137470996\">[latex]f\\left(x\\right)=\\frac{3}{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135435667\">\n<div id=\"fs-id1165135435669\">\n<p id=\"fs-id1165135208995\">[latex]f\\left(x\\right)={x}^{3}-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135541804\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135541807\">[latex]{f}^{-1}\\left(x\\right)={\\left(1+x\\right)}^{1\/3}[\/latex]<\/p>\n<p><span id=\"fs-id1165134057534\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142141\/CNX_Precalc_Figure_01_07_207.jpg\" alt=\"Graph of a cubic function and its inverse.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419353\">\n<div>\n<p id=\"fs-id1165135209711\">Find the inverse function of[latex]\\,f\\left(x\\right)=\\frac{1}{x-1}.\\,[\/latex]Use a graphing utility to find its domain and range. Write the domain and range in interval notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137592239\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135512365\">\n<div id=\"fs-id1165135512368\">\n<p id=\"fs-id1165135194429\">To convert from[latex]\\,x\\,[\/latex]degrees Celsius to[latex]\\,y\\,[\/latex]degrees Fahrenheit, we use the formula[latex]\\,f\\left(x\\right)=\\frac{9}{5}x+32.\\,[\/latex]Find the inverse function, if it exists, and explain its meaning.<\/p>\n<\/div>\n<div id=\"fs-id1165134164967\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134164969\">[latex]{f}^{-1}\\left(x\\right)=\\frac{5}{9}\\left(x-32\\right).\\,[\/latex]Given the Fahrenheit temperature,[latex]\\,x,\\,[\/latex]this formula allows you to calculate the Celsius temperature.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462831\">\n<div id=\"fs-id1165137462833\">\n<p id=\"fs-id1165137462835\">The circumference[latex]\\,C\\,[\/latex]of a circle is a function of its radius given by[latex]\\,C\\left(r\\right)=2\\pi r.\\,[\/latex]Express the radius of a circle as a function of its circumference. Call this function[latex]\\,r\\left(C\\right).\\,[\/latex]Find[latex]\\,r\\left(36\\pi \\right)\\,[\/latex]and interpret its meaning.<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135532476\">\n<p id=\"fs-id1165137645594\">A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time,[latex]\\,t,\\,[\/latex]in hours given by[latex]\\,d\\left(t\\right)=50t.\\,[\/latex]Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function[latex]\\,t\\left(d\\right).\\,[\/latex]Find[latex]\\,t\\left(180\\right)\\,[\/latex]and interpret its meaning.<\/p>\n<\/div>\n<div id=\"fs-id1165137766909\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137766912\">[latex]t\\left(d\\right)=\\frac{d}{50},\\,[\/latex][latex]t\\left(180\\right)=\\frac{180}{50}.\\,[\/latex]The time for the car to travel 180 miles is 3.6 hours.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135177582\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"eip-id1165135176875\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/55f2e8ec-a982-4586-9d48-a2f43d7b4107\">Functions and Function Notation<\/a><\/h4>\n<p id=\"fs-id1165137911358\">For the following exercises, determine whether the relation is a function.<\/p>\n<div id=\"fs-id1165137464074\">\n<div>[latex]\\left\\{\\left(a,b\\right),\\left(c,d\\right),\\left(e,d\\right)\\right\\}[\/latex]<\/div>\n<div id=\"fs-id1165137552980\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137552982\">function<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137728083\">\n<div id=\"fs-id1165137501972\">[latex]\\left\\{\\left(5,2\\right),\\left(6,1\\right),\\left(6,2\\right),\\left(4,8\\right)\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134569510\">\n<div id=\"fs-id1165134569512\">\n<p id=\"fs-id1165137446921\">[latex]{y}^{2}+4=x,\\,[\/latex]for[latex]\\,x\\,[\/latex]the independent variable and[latex]\\,y\\,[\/latex]the dependent variable<\/p>\n<\/div>\n<div id=\"fs-id1165134081333\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134081336\">not a function<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137464340\">\n<div id=\"fs-id1165137464343\">\n<p id=\"fs-id1165137828359\">Is the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_07_208\">(Figure)<\/a> a function?<\/p>\n<div id=\"Figure_01_07_208\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165135154408\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142202\/CNX_Precalc_Figure_01_07_208.jpg\" alt=\"Graph of a parabola.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137417004\">For the following exercises, evaluate the function at the indicated values:[latex]\\,\\,\\,f\\left(-3\\right);\\,\\,f\\left(2\\right);\\,\\,\\,f\\left(-a\\right);\\,\\,\\,-f\\left(a\\right);\\,\\,\\,f\\left(a+h\\right).[\/latex]<\/p>\n<div id=\"fs-id1165135186568\">\n<div id=\"fs-id1165135186571\">\n<p id=\"fs-id1165137728372\">[latex]f\\left(x\\right)=-2{x}^{2}+3x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137401744\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137446132\">[latex]f\\left(-3\\right)=-27;[\/latex][latex]f\\left(2\\right)=-2;[\/latex][latex]f\\left(-a\\right)=-2{a}^{2}-3a;[\/latex]<\/p>\n<p>[latex]-f\\left(a\\right)=2{a}^{2}-3a;[\/latex][latex]f\\left(a+h\\right)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}[\/latex]<\/details>\n<\/div>\n<div id=\"fs-id1165137651645\">\n<div id=\"fs-id1165137651647\">\n<p id=\"fs-id1165137651649\">[latex]f\\left(x\\right)=2|3x-1|[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135496646\">For the following exercises, determine whether the functions are one-to-one.<\/p>\n<div id=\"fs-id1165137605472\">\n<div id=\"fs-id1165137605474\">\n<p id=\"fs-id1165137580535\">[latex]f\\left(x\\right)=-3x+5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135181580\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135181582\">one-to-one<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731102\">\n<div id=\"fs-id1165137643347\">\n<p id=\"fs-id1165137643349\">[latex]f\\left(x\\right)=|x-3|[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137541296\">For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.<\/p>\n<div id=\"fs-id1165134324955\">\n<div id=\"fs-id1165134324957\"><span id=\"fs-id1165137416484\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142209\/CNX_Precalc_Figure_01_07_209.jpg\" alt=\"Graph of a cubic function.\" \/><\/span><\/div>\n<div id=\"fs-id1165137784950\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137784952\">function<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541845\">\n<div id=\"fs-id1165135541847\"><span id=\"fs-id1165137612052\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142211\/CNX_Precalc_Figure_01_07_210.jpg\" alt=\"Graph of a relation.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137653012\">\n<div id=\"fs-id1165137427968\"><span id=\"fs-id1165135168455\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142213\/CNX_Precalc_Figure_01_07_211.jpg\" alt=\"Graph of a relation.\" \/><\/span><\/div>\n<div id=\"fs-id1165137626904\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137626907\">function<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135255816\">For the following exercises, graph the functions.<\/p>\n<div id=\"fs-id1165137936642\">\n<div id=\"fs-id1165137936644\">\n<p id=\"fs-id1165137936646\">[latex]f\\left(x\\right)=|x+1|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137454708\">\n<div id=\"fs-id1165137454711\">\n<p id=\"fs-id1165137454713\">[latex]f\\left(x\\right)={x}^{2}-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137441682\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137641445\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142215\/CNX_Precalc_Figure_01_07_213.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"eip-860\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_01_07_215\">(Figure)<\/a> to approximate the values.<\/p>\n<div id=\"Figure_01_07_215\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137536184\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142225\/CNX_Precalc_Figure_01_07_215.jpg\" alt=\"Graph of a parabola.\" \/><\/span><\/div>\n<div id=\"fs-id1165137603626\">\n<div id=\"fs-id1165137603628\">\n<p id=\"fs-id1165135255923\">[latex]f\\left(2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134257596\">\n<div id=\"fs-id1165137579612\">\n<p id=\"fs-id1165137579614\">[latex]f\\left(-2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134269533\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134269535\">[latex]2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722552\">\n<div id=\"fs-id1165137722554\">\n<p id=\"fs-id1165137722556\">If[latex]\\,f\\left(x\\right)=-2,\\,[\/latex]then solve for[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823750\">\n<div id=\"fs-id1165137456896\">\n<p id=\"fs-id1165137456898\">If[latex]\\,f\\left(x\\right)=1,\\,[\/latex]then solve for[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137871525\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137462412\">[latex]x=-1.8\\text{ }[\/latex]or[latex]\\text{ or }x=1.8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137642827\">For the following exercises, use the function[latex]\\,h\\left(t\\right)=-16{t}^{2}+80t\\,[\/latex]to find the values in simplest form.<\/p>\n<div id=\"fs-id1165135160654\">\n<div id=\"fs-id1165135160656\">\n<p id=\"fs-id1165135160658\">[latex]\\frac{h\\left(2\\right)-h\\left(1\\right)}{2-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742766\">\n<div id=\"fs-id1165137742768\">\n<p id=\"fs-id1165137742770\">[latex]\\frac{h\\left(a\\right)-h\\left(1\\right)}{a-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135409755\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135409757\">[latex]\\frac{-64+80a-16{a}^{2}}{-1+a}=-16a+64[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165132944714\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/562c3737-a93d-458c-98c0-a04f442f13bd\">Domain and Range<\/a><\/h4>\n<p id=\"fs-id1165137667477\">For the following exercises, find the domain of each function, expressing answers using interval notation.<\/p>\n<div id=\"fs-id1165137667482\">\n<div id=\"fs-id1165137535162\">\n<p id=\"fs-id1165137535164\">[latex]f\\left(x\\right)=\\frac{2}{3x+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137648069\">\n<div id=\"fs-id1165137648071\">\n<p id=\"fs-id1165137936848\">[latex]f\\left(x\\right)=\\frac{x-3}{{x}^{2}-4x-12}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137552661\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137552663\">[latex]\\left(-\\infty ,-2\\right)\\cup \\left(-2,6\\right)\\cup \\left(6,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137933220\">\n<div id=\"fs-id1165131990652\">\n<p id=\"fs-id1165131990654\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135368485\">\n<div id=\"fs-id1165137665507\">\n<p id=\"fs-id1165137665510\">Graph this piecewise function:[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{l}x+1\\text{ }x<-2\\\\ -2x-3\\text{ }x\\ge -2\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133437266\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137803119\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142226\/CNX_Precalc_Figure_01_07_214.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165133183577\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f37919d7-b496-4e36-8196-431ae4733a64\">Rates of Change and Behavior of Graphs<\/a><\/h4>\n<p id=\"fs-id1165135161227\">For the following exercises, find the average rate of change of the functions from[latex]\\,x=1\\text{ to }x=2.[\/latex]<\/p>\n<div id=\"fs-id1165137541046\">\n<div id=\"fs-id1165137541048\">\n<p id=\"fs-id1165137541051\">[latex]f\\left(x\\right)=4x-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137664379\">\n<div id=\"fs-id1165137664381\">\n<p id=\"fs-id1165137664383\">[latex]f\\left(x\\right)=10{x}^{2}+x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134148505\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134148508\">[latex]31[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137807042\">\n<div id=\"fs-id1165137807044\">\n<p id=\"fs-id1165137601405\">[latex]f\\left(x\\right)=-\\frac{2}{{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137863445\">For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.<\/p>\n<div id=\"fs-id1165137527205\">\n<div id=\"fs-id1165137527207\"><span id=\"fs-id1165137734838\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142228\/CNX_Precalc_Figure_01_07_216.jpg\" alt=\"Graph of a parabola.\" \/><\/span><\/div>\n<div id=\"fs-id1165137911127\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137911128\">increasing[latex]\\,\\left(2,\\infty \\right);\\,[\/latex]<br \/>\ndecreasing[latex]\\,\\left(-\\infty ,2\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575929\">\n<div id=\"fs-id1165137768455\"><span id=\"fs-id1165135169395\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142240\/CNX_Precalc_Figure_01_07_217.jpg\" alt=\"Graph of a cubic function.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135153146\">\n<div id=\"fs-id1165135153148\"><span id=\"fs-id1165137640124\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142247\/CNX_Precalc_Figure_01_07_218.jpg\" alt=\"Graph of a function.\" \/><\/span><\/div>\n<div id=\"fs-id1165135439993\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137810327\">increasing[latex]\\text{}\\left(-3,1\\right);\\text{}[\/latex]constant[latex]\\,\\left(-\\infty ,-3\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727192\">\n<div id=\"fs-id1165137727194\">\n<p id=\"fs-id1165137727197\">Find the local minimum of the function graphed in <a class=\"autogenerated-content\" href=\"#fs-id1165137527205\">(Figure)<\/a>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135183100\">\n<div id=\"fs-id1165135183102\">\n<p id=\"fs-id1165135259506\">Find the local extrema for the function graphed in <a class=\"autogenerated-content\" href=\"#fs-id1165137575929\">(Figure)<\/a>.<\/p>\n<\/div>\n<div id=\"fs-id1165135259511\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137643970\">local minimum[latex]\\,\\left(-2,-3\\right);\\,[\/latex]local maximum[latex]\\,\\left(1,3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453948\">\n<div id=\"fs-id1165137453950\">\n<p id=\"fs-id1165137453952\">For the graph in <a class=\"autogenerated-content\" href=\"#Figure_01_07_219\">(Figure)<\/a>, the domain of the function is[latex]\\,\\left[-3,3\\right].[\/latex]The range is[latex]\\,\\left[-10,10\\right].\\,[\/latex]Find the absolute minimum of the function on this interval.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575004\">\n<div id=\"fs-id1165137575006\">\n<p id=\"fs-id1165137597151\">Find the absolute maximum of the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_01_07_219\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_219\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165137441761\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142300\/CNX_Precalc_Figure_01_07_219.jpg\" alt=\"Graph of a cubic function.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137619711\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137619713\">[latex]\\,\\left(-1.8,10\\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165131815810\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/8a4fc477-43da-4fc4-9ff0-19a7fae5a19d\">Composition of Functions<\/a><\/h4>\n<p id=\"fs-id1165132962099\">For the following exercises, find[latex]\\,\\left(f\\circ g\\right)\\left(x\\right)\\,[\/latex]and[latex]\\,\\left(g\\circ f\\right)\\left(x\\right)\\,[\/latex]for each pair of functions.<\/p>\n<div id=\"fs-id1165137535286\">\n<div id=\"fs-id1165137535288\">\n<p id=\"fs-id1165137653345\">[latex]f\\left(x\\right)=4-x,\\,g\\left(x\\right)=-4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137581604\">\n<div id=\"fs-id1165137581606\">\n<p id=\"fs-id1165137581608\">[latex]f\\left(x\\right)=3x+2,\\,g\\left(x\\right)=5-6x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137401055\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137401058\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=17-18x;\\,\\left(g\\circ f\\right)\\left(x\\right)=-7-18x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556913\">\n<div id=\"fs-id1165137556915\">\n<p id=\"fs-id1165137556917\">[latex]f\\left(x\\right)={x}^{2}+2x,\\,g\\left(x\\right)=5x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135508544\">\n<div id=\"fs-id1165135508547\">\n<p id=\"fs-id1165133337518\">[latex]f\\left(x\\right)=\\sqrt{x+2},\\text{ }g\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137506826\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137506828\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\sqrt{\\frac{1}{x}+2};\\,\\left(g\\circ f\\right)\\left(x\\right)=\\frac{1}{\\sqrt{x+2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134505601\">\n<div id=\"fs-id1165137437882\">\n<p id=\"fs-id1165137437884\">[latex]\\,f\\left(x\\right)=\\frac{x+3}{2},\\text{ }g\\left(x\\right)=\\sqrt{1-x}\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137411081\">For the following exercises, find[latex]\\,\\left(f\\circ g\\right)\\,[\/latex]and the domain for[latex]\\,\\left(f\\circ g\\right)\\left(x\\right)\\,[\/latex]for each pair of functions.<\/p>\n<div id=\"fs-id1165137851236\">\n<div id=\"fs-id1165137851238\">\n<p id=\"fs-id1165137851240\">[latex]f\\left(x\\right)=\\frac{x+1}{x+4},\\text{ }g\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137472983\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137588201\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\frac{1+x}{1+4x}, x\\ne 0, x\\ne -\\frac{1}{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407511\">\n<div id=\"fs-id1165135407513\">\n<p id=\"fs-id1165135407515\">[latex]f\\left(x\\right)=\\frac{1}{x+3},\\text{ }g\\left(x\\right)=\\frac{1}{x-9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137438445\">\n<div id=\"fs-id1165137596259\">\n<p id=\"fs-id1165137596261\">[latex]f\\left(x\\right)=\\frac{1}{x},\\text{ }g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134374025\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137456076\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\frac{1}{\\sqrt{x}},\\,x>0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135505012\">\n<div id=\"fs-id1165135505014\">\n<p id=\"fs-id1165137427123\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-1},\\text{ }g\\left(x\\right)=\\sqrt{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137734804\">For the following exercises, express each function[latex]\\,H\\,[\/latex]as a composition of two functions[latex]\\,f\\,[\/latex]and[latex]\\,g\\,[\/latex]where[latex]\\,H\\left(x\\right)=\\left(f\\circ g\\right)\\left(x\\right).[\/latex]<\/p>\n<div id=\"fs-id1165137448277\">\n<div id=\"fs-id1165137448279\">\n<p id=\"fs-id1165137837225\">[latex]H\\left(x\\right)=\\sqrt{\\frac{2x-1}{3x+4}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137697087\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137697090\">sample:[latex]\\,g\\left(x\\right)=\\frac{2x-1}{3x+4};\\,f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526679\">\n<div id=\"fs-id1165137526681\">\n<p id=\"fs-id1165137526683\">[latex]H\\left(x\\right)=\\frac{1}{{\\left(3{x}^{2}-4\\right)}^{-3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165134070725\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/5f6ff02a-1000-410d-b034-af26fbd86d0b\">Transformation of Functions<\/a><\/h4>\n<p id=\"fs-id1165137901175\">For the following exercises, sketch a graph of the given function.<\/p>\n<div id=\"fs-id1165137901178\">\n<div id=\"fs-id1165135439826\">\n<p id=\"fs-id1165135439828\">[latex]f\\left(x\\right)={\\left(x-3\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137665486\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135416583\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142314\/CNX_Precalc_Figure_01_07_220.jpg\" alt=\"Graph of f(x)\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419235\">\n<div id=\"fs-id1165137419237\">\n<p id=\"fs-id1165135358013\">[latex]f\\left(x\\right)={\\left(x+4\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137605818\">\n<div id=\"fs-id1165137597708\">\n<p id=\"fs-id1165137597710\">[latex]f\\left(x\\right)=\\sqrt{x}+5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137442696\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137768206\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142316\/CNX_Precalc_Figure_01_07_222.jpg\" alt=\"Graph of f(x)\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638654\">\n<div id=\"fs-id1165137638656\">\n<p id=\"fs-id1165134386603\">[latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137446722\">\n<div id=\"fs-id1165137565755\">\n<p id=\"fs-id1165137565757\">[latex]f\\left(x\\right)=\\sqrt[3]{-x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137771419\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135255841\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142321\/CNX_Precalc_Figure_01_07_224.jpg\" alt=\"Graph of f(x)\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137619832\">\n<div id=\"fs-id1165135421539\">\n<p id=\"fs-id1165135421541\">[latex]f\\left(x\\right)=5\\sqrt{-x}-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137793577\">\n<div id=\"fs-id1165137793579\">\n<p id=\"fs-id1165137793581\">[latex]f\\left(x\\right)=4\\left[|x-2|-6\\right][\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137651999\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137599946\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142330\/CNX_Precalc_Figure_01_07_226.jpg\" alt=\"Graph of f(x)\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628681\">\n<div id=\"fs-id1165137628683\">\n<p id=\"fs-id1165137628685\">[latex]f\\left(x\\right)=-{\\left(x+2\\right)}^{2}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137706346\">For the following exercises, sketch the graph of the function[latex]\\,g\\,[\/latex]if the graph of the function[latex]\\,f\\,[\/latex]is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_247\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_247\" class=\"medium\"><span id=\"fs-id1165137460564\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142332\/CNX_Precalc_Figure_01_07_247.jpg\" alt=\"Graph of f(x)\" \/><\/span><\/div>\n<div id=\"fs-id1165137736169\">\n<div id=\"fs-id1165137736171\">\n<p id=\"fs-id1165135152244\">[latex]g\\left(x\\right)=f\\left(x-1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137415538\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137442236\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142334\/CNX_Precalc_Figure_01_07_228.jpg\" alt=\"Graph of a half circle.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135543526\">\n<div id=\"fs-id1165137742527\">\n<p id=\"fs-id1165137742529\">[latex]g\\left(x\\right)=3f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137626728\">For the following exercises, write the equation for the standard function represented by each of the graphs below.<\/p>\n<div id=\"fs-id1165137562807\">\n<div id=\"fs-id1165137562809\"><span id=\"fs-id1165137446970\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142336\/CNX_Precalc_Figure_01_07_230.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/div>\n<div id=\"fs-id1165137761321\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137937561\">[latex]f\\left(x\\right)=|x-3|[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137570135\">\n<div id=\"fs-id1165137570137\"><span id=\"fs-id1165135524458\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142347\/CNX_Precalc_Figure_01_07_231.jpg\" alt=\"Graph of a half circle.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165135496278\">For the following exercises, determine whether each function below is even, odd, or neither.<\/p>\n<div id=\"fs-id1165135496282\">\n<div id=\"fs-id1165137416232\">\n<p id=\"fs-id1165137416234\">[latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137812584\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137812586\">even<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137394734\">\n<div id=\"fs-id1165137394736\">\n<p id=\"fs-id1165134258620\">[latex]g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176486\">\n<div id=\"fs-id1165135176488\">\n<p id=\"fs-id1165135176491\">[latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137598143\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137598145\">odd<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137565963\">For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.<\/p>\n<div id=\"fs-id1165137565968\">\n<div id=\"fs-id1165134148336\"><span id=\"fs-id1165137534178\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142354\/CNX_Precalc_Figure_01_07_232.jpg\" alt=\"Graph of a parabola.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137643328\">\n<div id=\"fs-id1165137643331\"><span id=\"fs-id1165137836434\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142402\/CNX_Precalc_Figure_01_07_233.jpg\" alt=\"Graph of a parabola.\" \/><\/span><\/div>\n<div id=\"fs-id1165135344925\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135545915\">even<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135471211\">\n<div id=\"fs-id1165135471213\"><span id=\"fs-id1165135538487\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142408\/CNX_Precalc_Figure_01_07_234.jpg\" alt=\"Graph of a cubic function.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165135691363\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/2e387575-c04f-40e1-8895-195affae8fdb\">Absolute Value Functions<\/a><\/h4>\n<p id=\"fs-id1165135203710\">For the following exercises, write an equation for the transformation of[latex]\\,f\\left(x\\right)=|x|.[\/latex]<\/p>\n<div id=\"fs-id1165137725549\">\n<div id=\"fs-id1165137470719\"><span id=\"fs-id1165137597393\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142411\/CNX_Precalc_Figure_01_07_235.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/div>\n<div id=\"fs-id1165137635245\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137635247\">[latex]f\\left(x\\right)=\\frac{1}{2}|x+2|+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836804\">\n<div id=\"fs-id1165137836807\"><span id=\"fs-id1165137849182\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142412\/CNX_Precalc_Figure_01_07_236.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137737661\">\n<div id=\"fs-id1165134224080\"><span id=\"fs-id1165137656022\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142413\/CNX_Precalc_Figure_01_07_237.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/div>\n<div id=\"fs-id1165135181351\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135181354\">[latex]f\\left(x\\right)=-3|x-3|+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137415922\">For the following exercises, graph the absolute value function.<\/p>\n<div id=\"fs-id1165137426335\">\n<div id=\"fs-id1165137426337\">\n<p id=\"fs-id1165137426339\">[latex]f\\left(x\\right)=|x-5|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134058433\">\n<div id=\"fs-id1165134058435\">\n<p id=\"fs-id1165134058437\">[latex]f\\left(x\\right)=-|x-3|[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134151182\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135256168\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142421\/CNX_Precalc_Figure_01_07_239.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673910\">\n<div id=\"fs-id1165135708035\">\n<p id=\"fs-id1165135708037\">[latex]f\\left(x\\right)=|2x-4|[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-159\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f592aad0-19d8-42d6-94b9-086bdd84c2b5\">Inverse Functions<\/a><\/h4>\n<p id=\"eip-id1964739\">For the following exercises, find[latex]\\text{ }{f}^{-1}\\left(x\\right)\\text{ }[\/latex]for each function.<\/p>\n<div>\n<div id=\"fs-id1165135255940\">\n<p id=\"fs-id1165135255942\">[latex]f\\left(x\\right)=9+10x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"eip-553\">\n<div id=\"eip-75\">\n<p id=\"eip-735\">[latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\n<\/div>\n<div id=\"eip-813\">\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{f}^{-1}\\left(x\\right)=\\frac{-2x}{x-1}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-293\">For the following exercise, find a domain on which the function[latex]\\text{ }f\\text{ }[\/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\\text{ }f\\text{ }[\/latex]restricted to that domain.<\/p>\n<div id=\"eip-117\">\n<div id=\"eip-537\">\n<p id=\"eip-385\">[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"eip-98\">\n<div id=\"eip-578\">\n<p id=\"eip-395\">Given [latex]f\\left(x\\right)={x}^{3}-5[\/latex] and [latex]g\\left(x\\right)=\\sqrt[3]{x+5}:[\/latex]<\/p>\n<ol id=\"eip-id1165134205830\" type=\"a\">\n<li>Find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right).[\/latex]<\/li>\n<li>What does the answer tell us about the relationship between [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"eip-id1165134350154\" type=\"a\">\n<li>[latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x.[\/latex]<\/li>\n<li>This tells us that [latex]f[\/latex] and [latex]g[\/latex] are inverse functions<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"eip-712\">For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n<div id=\"eip-72\">\n<div id=\"eip-808\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/div>\n<div id=\"eip-845\">\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"eip-366\">The function is one-to-one.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142428\/CNX_Precalc_Figure_01_07_248.jpg\" alt=\"\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-642\">\n<div id=\"eip-85\">\n<p id=\"eip-256\">[latex]f\\left(x\\right)=-3{x}^{2}+x[\/latex]<\/p>\n<\/div>\n<div id=\"eip-id2171835\">\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"eip-id2171848\">The function is not one-to-one.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142431\/CNX_Precalc_Figure_01_07_249.jpg\" alt=\"\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-491\">\n<div id=\"eip-368\">\n<p>If [latex]f\\left(5\\right)=2,[\/latex] find [latex]{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<p id=\"eip-806\">\n<details>\n<summary>Show Solution<\/summary>\n<\/p>\n<div id=\"eip-491\">\n<div id=\"eip-627\">\n<p id=\"eip-520\">[latex]5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-601\">\n<div id=\"eip-650\">\n<p id=\"eip-811\">If [latex]f\\left(1\\right)=4,[\/latex] find [latex]{f}^{-1}\\left(4\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137455205\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<p id=\"fs-id1165137590417\">For the following exercises, determine whether each of the following relations is a function.<\/p>\n<div id=\"fs-id1165137474063\">\n<div id=\"fs-id1165137474065\">\n<p id=\"fs-id1165137474067\">[latex]y=2x+8[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134572580\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134572582\">The relation is a function.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736465\">\n<div id=\"fs-id1165137736467\">\n<p id=\"fs-id1165137832014\">[latex]\\left\\{\\left(2,1\\right),\\left(3,2\\right),\\left(-1,1\\right),\\left(0,-2\\right)\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137734462\">For the following exercises, evaluate the function[latex]\\,f\\left(x\\right)=-3{x}^{2}+2x\\,[\/latex]<br \/>\nat the given input.<\/p>\n<div>\n<div>[latex]f\\left(-2\\right)[\/latex]<\/div>\n<div id=\"fs-id1165137472994\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137472996\">\u221216<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135182930\">\n<div id=\"fs-id1165137535259\">\n<p id=\"fs-id1165137535261\">[latex]\\,f\\left(a\\right)\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541955\">\n<div id=\"fs-id1165137634239\">\n<p id=\"fs-id1165137634241\">Show that the function[latex]\\,f\\left(x\\right)=-2{\\left(x-1\\right)}^{2}+3\\,[\/latex]is not one-to-one.<\/p>\n<\/div>\n<div id=\"fs-id1165135486043\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135486045\">The graph is a parabola and the graph fails the horizontal line test.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573629\">\n<div id=\"fs-id1165137400957\">\n<p id=\"fs-id1165137400960\">Write the domain of the function[latex]\\,f\\left(x\\right)=\\sqrt{3-x}\\,[\/latex]in interval notation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137405680\">\n<div id=\"fs-id1165137405682\">\n<p id=\"fs-id1165135202594\">Given[latex]\\,f\\left(x\\right)=2{x}^{2}-5x,\\,[\/latex]find[latex]f\\left(a+1\\right)-f\\left(1\\right)\\,[\/latex]in simplest form.<\/p>\n<\/div>\n<div id=\"fs-id1165137455933\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137812635\">[latex]2{a}^{2}-a[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137724050\">\n<div id=\"fs-id1165137724052\">\n<p id=\"fs-id1165137724054\">Graph the function[latex]f\\left(x\\right)=\\left\\{\\begin{array}{cc}x+1\\text{ if}& -2<x<3\\\\ \\text{ }-x\\text{ if }& x\\ge 3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810666\">\n<div id=\"fs-id1165137810669\">\n<p id=\"fs-id1165137810671\">Find the average rate of change of the function[latex]\\,f\\left(x\\right)=3-2{x}^{2}+x\\,[\/latex]by finding[latex]\\,\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}\\,[\/latex]in simplest form.<\/p>\n<\/div>\n<div id=\"fs-id1165135149027\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135149029\">[latex]-2\\left(a+b\\right)+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137730928\">For the following exercises, use the functions[latex]\\,f\\left(x\\right)=3-2{x}^{2}+x\\text{ and }g\\left(x\\right)=\\sqrt{x}\\,[\/latex]to find the composite functions.<\/p>\n<div id=\"fs-id1165135332782\">\n<div id=\"fs-id1165137471389\">\n<p id=\"fs-id1165137471391\">[latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040544\">\n<div id=\"fs-id1165134040546\">\n<p id=\"fs-id1165134040549\">[latex]\\left(g\\circ f\\right)\\left(1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137572557\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137572559\">[latex]\\sqrt{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433228\">\n<div id=\"fs-id1165137433230\">\n<p id=\"fs-id1165137863995\">Express[latex]\\,H\\left(x\\right)=\\sqrt[3]{5{x}^{2}-3x}\\,[\/latex]as a composition of two functions,[latex]\\,f\\,[\/latex]and[latex]\\,g,\\,[\/latex]where[latex]\\,\\left(f\\circ g\\right)\\left(x\\right)=H\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137849036\">For the following exercises, graph the functions by translating, stretching, and\/or compressing a toolkit function.<\/p>\n<div id=\"fs-id1165137724075\">\n<div id=\"fs-id1165137724077\">\n<p id=\"fs-id1165137714976\">[latex]f\\left(x\\right)=\\sqrt{x+6}-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137400919\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137454070\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142433\/CNX_Precalc_Figure_01_07_242.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137748677\">\n<div id=\"fs-id1165137748679\">\n<p id=\"fs-id1165137748681\">[latex]f\\left(x\\right)=\\frac{1}{x+2}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135245866\">For the following exercises, determine whether the functions are even, odd, or neither.<\/p>\n<div id=\"fs-id1165137451069\">\n<div id=\"fs-id1165137451071\">\n<p id=\"fs-id1165137423842\">[latex]f\\left(x\\right)=-\\frac{5}{{x}^{2}}+9{x}^{6}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137610712\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137610714\">[latex]\\text{even}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542465\">\n<div id=\"fs-id1165134250812\">[latex]f\\left(x\\right)=-\\frac{5}{{x}^{3}}+9{x}^{5}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137770368\">\n<div id=\"fs-id1165135536626\">\n<p id=\"fs-id1165135536628\">[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137540956\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137534626\">[latex]\\text{odd}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137933901\">\n<div id=\"fs-id1165137933903\">\n<p id=\"fs-id1165137933906\">Graph the absolute value function[latex]\\,f\\left(x\\right)=-2|x-1|+3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137675389\">For the following exercises, find the inverse of the function.<\/p>\n<div id=\"fs-id1165137675392\">\n<div id=\"fs-id1165134389015\">\n<p id=\"fs-id1165134389017\">[latex]f\\left(x\\right)=3x-5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135449688\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135449690\">[latex]{f}^{-1}\\left(x\\right)=\\frac{x+5}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135195662\">\n<div id=\"fs-id1165135195664\">\n<p id=\"fs-id1165135195667\">[latex]f\\left(x\\right)=\\frac{4}{x+7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137727655\">For the following exercises, use the graph of[latex]\\,g\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_245\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_245\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165137456874\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142439\/CNX_Precalc_Figure_01_07_245.jpg\" alt=\"Graph of a cubic function.\" \/><\/span><\/div>\n<div id=\"fs-id1165137589849\">\n<div id=\"fs-id1165137589851\">\n<p id=\"fs-id1165137589853\">On what intervals is the function increasing?<\/p>\n<\/div>\n<div id=\"fs-id1165137661790\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137661792\">[latex]\\left(-\\infty ,-1.1\\right)\\text{ and }\\left(1.1,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137942391\">\n<div id=\"fs-id1165137942393\">\n<p id=\"fs-id1165137942395\">On what intervals is the function decreasing?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137463490\">\n<div id=\"fs-id1165134558010\">\n<p id=\"fs-id1165134558012\">Approximate the local minimum of the function. Express the answer as an ordered pair.<\/p>\n<\/div>\n<div id=\"fs-id1165137619563\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137619565\">[latex]\\left(1.1,-0.9\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135321930\">\n<div id=\"fs-id1165137679083\">\n<p id=\"fs-id1165137679085\">Approximate the local maximum of the function. Express the answer as an ordered pair.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137933195\">For the following exercises, use the graph of the piecewise function shown in <a class=\"autogenerated-content\" href=\"#Figure_01_07_246\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_07_246\" class=\"small wp-caption aligncenter\"><span id=\"fs-id1165135529085\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19142444\/CNX_Precalc_Figure_01_07_246.jpg\" alt=\"Graph of absolute function and step function.\" \/><\/span><\/div>\n<div id=\"fs-id1165135517182\">\n<div id=\"fs-id1165135517184\">\n<p id=\"fs-id1165135517186\">Find[latex]\\,f\\left(2\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137762681\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137762683\">[latex]f\\left(2\\right)=2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834540\">\n<div id=\"fs-id1165137443779\">\n<p id=\"fs-id1165137443781\">Find[latex]\\,f\\left(-2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135168371\">\n<div id=\"fs-id1165135168373\">\n<p id=\"fs-id1165137556971\">Write an equation for the piecewise function.<\/p>\n<\/div>\n<div id=\"fs-id1165137556975\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137726425\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}|x|\\,\\,\\,\\text{if}\\,\\,x\\le 2\\\\ 3\\,\\,\\,\\,\\,\\text{if}\\,\\,x>2\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137696459\">For the following exercises, use the values listed in <a class=\"autogenerated-content\" href=\"#Table_01_07_07\">(Figure)<\/a>.<\/p>\n<table id=\"Table_01_07_07\" summary=\"..\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]F\\left(x\\right)[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>13<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165135149302\">\n<div id=\"fs-id1165135149304\">\n<p id=\"fs-id1165135149306\">Find[latex]\\,F\\left(6\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135616339\">\n<div id=\"fs-id1165135616341\">\n<p id=\"fs-id1165135616343\">Solve the equation[latex]\\,F\\left(x\\right)=5.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137676959\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137676962\">[latex]x=2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836651\">\n<div id=\"fs-id1165137836653\">\n<p id=\"fs-id1165135256131\">Is the graph increasing or decreasing on its domain?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137850361\">\n<div id=\"fs-id1165137850363\">\n<p id=\"fs-id1165137850365\">Is the function represented by the graph one-to-one?<\/p>\n<\/div>\n<div id=\"fs-id1165137736528\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137736530\">yes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447422\">\n<div id=\"fs-id1165137447424\">\n<p id=\"fs-id1165137469729\">Find[latex]\\,{F}^{-1}\\left(15\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549524\">\n<div id=\"fs-id1165137549526\">\n<p id=\"fs-id1165137758292\">Given[latex]\\,f\\left(x\\right)=-2x+11,\\,[\/latex]find[latex]\\,{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137851399\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137851402\">[latex]{f}^{-1}\\left(x\\right)=-\\frac{x-11}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137441703\">\n<dt>inverse function<\/dt>\n<dd id=\"fs-id1165137441708\">for any one-to-one function[latex]\\,f\\left(x\\right),\\,[\/latex]the inverse is a function[latex]\\,{f}^{-1}\\left(x\\right)\\,[\/latex]such that[latex]\\,{f}^{-1}\\left(f\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f;\\,[\/latex]this also implies that[latex]\\,f\\left({f}^{-1}\\left(x\\right)\\right)=x\\,[\/latex]for all[latex]\\,x\\,[\/latex]in the domain of[latex]\\,{f}^{-1}[\/latex]<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":8,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-65","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/65","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/65\/revisions"}],"predecessor-version":[{"id":66,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/65\/revisions\/66"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/50"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/65\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=65"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=65"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=65"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=65"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}