{"id":55,"date":"2019-08-20T17:01:44","date_gmt":"2019-08-20T21:01:44","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/domain-and-range\/"},"modified":"2022-06-01T10:39:22","modified_gmt":"2022-06-01T14:39:22","slug":"domain-and-range","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/domain-and-range\/","title":{"raw":"Domain and Range","rendered":"Domain and Range"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Find the domain of a function defined by an equation.<\/li>\n \t<li>Graph piecewise-defined functions.<\/li>\n<\/ul>\n<\/div>\nIf you\u2019re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time\u2014<em>I am Legend<\/em>, <em>Hannibal<\/em>, <em>The Ring<\/em>, <em>The Grudge<\/em>, and <em>The Conjuring<\/em>. <a class=\"autogenerated-content\" href=\"#Figure_01_02_001\">(Figure)<\/a> shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <span class=\"no-emphasis\">domain<\/span> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/cnx.org\/resources\/f29850a56c5a5ec5355fe59ed2e7ce1a3bb85627\/CNX_Precalc_Figure_01_02_001.jpg\" alt=\"Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year\" width=\"975\" height=\"402\"> <strong>Figure 1.<\/strong> Based on data compiled by www.the-numbers.com.[\/caption]\n\n[footnote]The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014[\/footnote]\n<div id=\"fs-id1165135193832\" class=\"bc-section section\">\n<h3>Finding the Domain of a Function Defined by an Equation<\/h3>\n<p id=\"fs-id1165135445896\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/55f2e8ec-a982-4586-9d48-a2f43d7b4107\">Functions and Function Notation<\/a>, we were introduced to the concepts of <span class=\"no-emphasis\">domain and range<\/span>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \u201cholding area\u201d that contains \u201craw materials\u201d for a \u201cfunction machine\u201d and the range as another \u201cholding area\u201d for the machine\u2019s products. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_002\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_02_002\" 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\/19134739\/CNX_Precalc_Figure_01_02_002.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137761714\">We can write the <span class=\"no-emphasis\">domain and range<\/span> in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write[latex]\\,\\left(0,\\text{ }100\\right].\\,[\/latex]We will discuss interval notation in greater detail later.<\/p>\n<p id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\n\n<ul id=\"fs-id1165135673417\">\n \t<li>The smallest number from the interval is written first.<\/li>\n \t<li>The largest number in the interval is written second, following a comma.<\/li>\n \t<li>Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.<\/li>\n \t<li>Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.<\/li>\n<\/ul>\n<p id=\"fs-id1165137807384\">See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_029\">(Figure)<\/a> for a summary of interval notation.<\/p>\n\n<div id=\"Figure_01_02_029\" 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\/19134741\/CNX_Precalc_Figure_01_02_029n.jpg\" alt=\"Summary of interval notation.\" width=\"975\" height=\"905\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137661548\">\n<div id=\"fs-id1165137772018\">\n<h3>Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\n<p id=\"fs-id1165137920768\">Find the <span class=\"no-emphasis\">domain<\/span> of the following function:[latex]\\,\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex].<\/p>\n\n<\/div>\n<div id=\"fs-id1165135329797\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135329797\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135329797\"]\n<p id=\"fs-id1165135508343\">First identify the input values. The input value is the first coordinate in an <span class=\"no-emphasis\">ordered pair<\/span>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\n\n<div id=\"fs-id1165137451888\" class=\"unnumbered aligncenter\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137569901\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165135333722\">\n<div id=\"fs-id1165137852040\">\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\n<p id=\"fs-id1165137466017\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137501477\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137501477\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137501477\"]\n<p id=\"fs-id1165137704712\">[latex]\\left\\{-5,\\,0,\\,5,\\,10,\\,15\\right\\}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134225655\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134355557\"><strong>Given a function written in equation form, find the domain.<\/strong><\/p>\n\n<ol id=\"fs-id1165134187286\" type=\"1\">\n \t<li>Identify the input values.<\/li>\n \t<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\n \t<li>Write the domain in interval form, if possible.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137767649\">\n<div id=\"fs-id1165137761307\">\n<h3>Finding the Domain of a Function<\/h3>\n<p id=\"fs-id1165137645656\">Find the domain of the function[latex]\\,f\\left(x\\right)={x}^{2}-1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135684349\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135684349\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135684349\"]\n<p id=\"fs-id1165137594433\">The input value, shown by the variable[latex]\\,x\\,[\/latex]in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.<\/p>\n<p id=\"fs-id1165135309759\">In interval form, the domain of[latex]\\,f\\,[\/latex]is[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135639906\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137733850\">\n<div id=\"fs-id1165137871971\">\n<p id=\"fs-id1165137871972\">Find the domain of the function:[latex]\\,f\\left(x\\right)=5-x+{x}^{3}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137809848\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137809848\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137809848\"]\n<p id=\"fs-id1165137809849\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417188\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137473617\"><strong>Given a function written in an equation form that includes a fraction, find the domain.<\/strong><\/p>\n\n<ol id=\"fs-id1165137463251\" type=\"1\">\n \t<li>Identify the input values.<\/li>\n \t<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for[latex]\\,x\\,[\/latex]. If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\n \t<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137722406\">\n<div id=\"fs-id1165135484119\">\n<h3>Finding the Domain of a Function Involving a Denominator<\/h3>\n<p id=\"fs-id1165137647592\">Find the <span class=\"no-emphasis\">domain<\/span> of the function[latex]\\,f\\left(x\\right)=\\frac{x+1}{2-x}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135641743\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135641743\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135641743\"]\n<p id=\"fs-id1165137565519\">When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1165137736620\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 2-x&amp; =&amp; 0\\hfill \\\\ \\hfill -x&amp; =&amp; -2\\hfill \\\\ \\hfill x&amp; =&amp; 2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135192763\">Now, we will exclude 2 from the domain. The answers are all real numbers where[latex]\\,x&lt;2\\,[\/latex]or[latex]\\,x&gt;2\\,[\/latex]as shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Image_01_02_028\">(Figure)<\/a>. We can use a symbol known as the union,[latex]\\,\\cup ,[\/latex]to combine the two sets. In interval notation, we write the solution:[latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right).[\/latex]<\/p>\n\n<div id=\"Image_01_02_028\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19134744\/CNX_Precalc_Figure_01_02_028n.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\"> <strong>Figure 4. <\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133349280\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137437630\">\n<div id=\"fs-id1165137771815\">\n<p id=\"fs-id1165137442339\">Find the domain of the function:[latex]\\,f\\left(x\\right)=\\frac{1+4x}{2x-1}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137436024\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137436024\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137436024\"]\n<p id=\"fs-id1165135186314\">[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135527005\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137733733\"><strong>Given a function written in equation form including an even root, find the domain.<\/strong><\/p>\n\n<ol id=\"fs-id1165137820030\" type=\"1\">\n \t<li>Identify the input values.<\/li>\n \t<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for[latex]\\,x.[\/latex]<\/li>\n \t<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135160109\">\n<div id=\"fs-id1165137735699\">\n<h3>Finding the Domain of a Function with an Even Root<\/h3>\n<p id=\"fs-id1165137466144\">Find the <span class=\"no-emphasis\">domain<\/span> of the function[latex]\\,f\\left(x\\right)=\\sqrt{7-x}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137451129\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137451129\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137451129\"]\n<p id=\"fs-id1165137453224\">When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.<\/p>\n<p id=\"fs-id1165137749755\">Set the radicand greater than or equal to zero and solve for[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1165137727831\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 7-x&amp; \\ge &amp; 0\\hfill \\\\ \\hfill -x&amp; \\ge &amp; -7\\hfill \\\\ \\hfill x&amp; \\le &amp; 7\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137422794\">Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to[latex]\\,7,\\,[\/latex]or[latex]\\,\\left(-\\infty ,7\\right].[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737842\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137933139\">\n<div id=\"fs-id1165137933140\">\n<p id=\"fs-id1165137452448\">Find the domain of the function[latex]\\,f\\left(x\\right)=\\sqrt{5+2x}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137832331\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137832331\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137832331\"]\n<p id=\"fs-id1165137832332\">[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134328219\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137659456\"><strong>Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\n<p id=\"fs-id1165137937737\"><em>Yes. For example, the function[latex]\\,f\\left(x\\right)=-\\frac{1}{\\sqrt{x}}\\,[\/latex]has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137677916\" class=\"bc-section section\">\n<h3>Using Notations to Specify Domain and Range<\/h3>\n<p id=\"fs-id1165137410091\">In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example,[latex]\\,\\left\\{x|10\\le x&lt;30\\right\\}\\,[\/latex]describes the behavior of[latex]\\,x\\,[\/latex]in set-builder notation. The braces[latex]\\,\\left\\{\\right\\}\\,[\/latex]are read as \u201cthe set of,\u201d and the vertical bar | is read as \u201csuch that,\u201d so we would read[latex]\\,\\left\\{x|10\\le x&lt;30\\right\\}\\,[\/latex]as \u201cthe set of <em>x<\/em>-values such that 10 is less than or equal to[latex]\\,x,\\,[\/latex]and[latex]\\,x\\,[\/latex]is less than 30.\u201d<\/p>\n<p id=\"fs-id1165135207589\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_003\">(Figure)<\/a> compares inequality notation, set-builder notation, and interval notation.<\/p>\n\n<div id=\"Figure_01_02_003\" 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\/19134751\/CNX_Precalc_Figure_01_02_003.jpg\" alt=\"Summary of notations for inequalities, set-builder, and intervals.\" width=\"975\" height=\"692\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \u201cor.\u201d As we saw in earlier examples, we use the union symbol,[latex]\\,\\cup ,[\/latex]to combine two unconnected intervals. For example, the union of the sets[latex]\\left\\{2,3,5\\right\\}\\,[\/latex]\nand[latex]\\,\\left\\{4,6\\right\\}\\,[\/latex]\nis the set[latex]\\,\\left\\{2,3,4,5,6\\right\\}.\\,[\/latex]It is the set of all elements that belong to one <em>or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is<\/p>\n\n<div id=\"fs-id1165135311695\" class=\"unnumbered aligncenter\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/div>\n<div id=\"fs-id1165137641795\" class=\"textbox key-takeaways\">\n<h3>Set-Builder Notation and Interval Notation<\/h3>\n<p id=\"fs-id1165137663670\"><strong>Set-builder notation <\/strong>is a method of specifying a set of elements that satisfy a certain condition. It takes the form[latex]\\left\\{x|\\,\\text{statement about }x\\right\\}\\,[\/latex]which is read as, \u201cthe set of all[latex]\\,x\\,[\/latex]such that the statement about[latex]\\,x\\,[\/latex]is true.\u201d For example,<\/p>\n\n<div id=\"fs-id1165137543047\" class=\"unnumbered aligncenter\">[latex]\\left\\{x|4&lt;x\\le 12\\right\\}[\/latex]<\/div>\n<p id=\"fs-id1165135190272\"><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,<\/p>\n\n<div id=\"fs-id1165137443063\" class=\"unnumbered aligncenter\">[latex]\\left(4,12\\right][\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137805770\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137423878\"><strong>Given a line graph, describe the set of values using interval notation.<\/strong><\/p>\n\n<ol id=\"fs-id1165134032280\" type=\"1\">\n \t<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\n \t<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\n \t<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\n \t<li>Use the union symbol[latex]\\,\\cup \\,[\/latex]to combine all intervals into one set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165134342702\">\n<div id=\"fs-id1165137803670\">\n<h3>Describing Sets on the Real-Number Line<\/h3>\n<p id=\"fs-id1165137592069\">Describe the intervals of values shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_004\">(Figure)<\/a> using inequality notation, set-builder notation, and interval notation.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/13c74d3bc393a72d003d8fa46c769c591786bc87\/CNX_Precalc_Figure_01_02_004.jpg\" alt=\"Line graph of 1<=x<=3 and 5<x.\" width=\"487\" height=\"50\"> <strong>Figure 6.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135412904\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135412904\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135412904\"]\n<p id=\"fs-id1165135412905\">To describe the values,[latex]\\,x,\\,[\/latex]included in the intervals shown, we would say, \u201c[latex]x\\,[\/latex]is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\u201d<\/p>\n\n<table id=\"fs-id1165137447518\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Inequality<\/strong><\/td>\n<td>[latex]1\\le x\\le 3\\,\\text{or}\\,x&gt;5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Set-builder notation<\/strong><\/td>\n<td>[latex]\\left\\{x|1\\le x\\le 3\\,\\text{or}\\,x&gt;5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Interval notation<\/strong><\/td>\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135500794\">Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137779165\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135175087\">\n<p id=\"fs-id1165135341412\">Given <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_005\">(Figure)<\/a>, specify the graphed set in<\/p>\n\n<ol id=\"fs-id1165137595582\" type=\"a\">\n \t<li>words<\/li>\n \t<li>set-builder notation<\/li>\n \t<li>interval notation<\/li>\n<\/ol>\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/eba2f61d5fb32eca2dd57b7de1a1e57511a15f6a\/CNX_Precalc_Figure_01_02_005.jpg\" alt=\"Line graph of -2<=x, -1<=x<3.\" width=\"487\" height=\"50\"> <strong>Figure 7.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135209390\" class=\"solution textbox shaded\">\n<ol id=\"fs-id1165135528963\" type=\"a\">\n \t<li>values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3;<\/li>\n \t<li>[latex]\\left\\{x|x\\le -2\\,\\text{or}\\,-1\\le x&lt;3\\right\\}[\/latex]\n;<\/li>\n \t<li>[latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653855\" class=\"bc-section section\">\n<h3>Finding Domain and Range from Graphs<\/h3>\n<p id=\"fs-id1165135161404\">Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em>x<\/em>-axis. The range is the set of possible output values, which are shown on the <em>y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_006\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_02_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\/19134810\/CNX_Precalc_Figure_01_02_006.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\"> <strong>Figure 8.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137597994\">We can observe that the graph extends horizontally from[latex]\\,-5\\,[\/latex]to the right without bound, so the domain is[latex]\\,\\left[-5,\\infty \\right).\\,\\,[\/latex]The vertical extent of the graph is all range values[latex]\\,5\\,[\/latex]and below, so the range is[latex]\\,\\left(\\mathrm{-\\infty },5\\right].\\,[\/latex]Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137561401\">\n<div id=\"fs-id1165137599824\">\n<h3>Finding Domain and Range from a Graph<\/h3>\n<p id=\"fs-id1165135187604\">Find the domain and range of the function[latex]\\,f\\,[\/latex]\nwhose graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_007\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_02_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\/19134812\/CNX_Precalc_Figure_01_02_007.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575085\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137575085\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137575085\"]\n<p id=\"fs-id1165137768165\">We can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of[latex]\\,f\\,[\/latex]\nis[latex]\\,\\left(-3,1\\right].[\/latex]<\/p>\n<p id=\"fs-id1165131968670\">The vertical extent of the graph is 0 to \u20134, so the range is[latex]\\,\\left[-4,0\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_008\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_02_008\" 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\/19134818\/CNX_Precalc_Figure_01_02_008.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\"> <strong>Figure 10.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165134182686\">\n<div id=\"fs-id1165137461643\">\n<h3>Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p id=\"fs-id1165137443324\">Find the domain and range of the function[latex]\\,f\\,[\/latex]whose graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_009\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_02_009\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"489\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19134821\/CNX_Precalc_Figure_01_02_009.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\"> <strong>Figure 11. <\/strong>(credit: modification of work by the U.S. Energy Information Administration)[\/caption]\n\n[footnote]http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.[\/footnote]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137444311\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137444311\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137444311\"]\n<p id=\"fs-id1165137476085\">The input quantity along the horizontal axis is \u201cyears,\u201d which we represent with the variable[latex]\\,t\\,[\/latex]for time. The output quantity is \u201cthousands of barrels of oil per day,\u201d which we represent with the variable[latex]\\,b\\,[\/latex]for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as[latex]\\,1973\\le t\\le 2008\\,[\/latex]and the range as approximately[latex]\\,180\\le b\\le 2010.[\/latex]<\/p>\n<p id=\"fs-id1165137747998\">In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545972\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137644581\">\n<p id=\"fs-id1165137644582\">Given <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_010\">(Figure)<\/a>, identify the domain and range using interval notation.<\/p>\n\n<div id=\"Figure_01_02_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\/19134828\/CNX_Precalc_Figure_01_02_010.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705252\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137705252\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137705252\"]\n<p id=\"fs-id1165134079741\">domain =[1950,2002] range = [47,000,000,89,000,000]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434590\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137812796\"><strong>Can a function\u2019s domain and range be the same?<\/strong><\/p>\n<p id=\"fs-id1165137433394\"><em>Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134384565\" class=\"bc-section section\">\n<h3>Finding Domains and Ranges of the Toolkit Functions<\/h3>\n<p id=\"fs-id1165137419914\">We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n\n<div id=\"Figure_01_02_011\" 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\/19134830\/CNX_Precalc_Figure_01_02_011.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\"> <strong>Figure 13. <\/strong>For the <strong>constant function<\/strong>[latex]\\,f\\left(x\\right)=c,\\,[\/latex]the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant[latex]\\,c,\\,[\/latex]so the range is the set[latex]\\,\\left\\{c\\right\\}\\,[\/latex]that contains this single element. In interval notation, this is written as[latex]\\,\\left[c,c\\right],\\,[\/latex]the interval that both begins and ends with[latex]\\,c.[\/latex][\/caption]<\/div>\n<div id=\"Figure_01_02_012\" 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\/19134832\/CNX_Precalc_Figure_01_02_012.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\"> <strong>Figure 14.<\/strong> For the <strong>identity function\u2009<\/strong>f(x)=x, there is no restriction on\u2009x. Both the domain and range are the set of all real numbers.[\/caption]\n\n<\/div>\n<div id=\"Figure_01_02_013\" 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\/19134844\/CNX_Precalc_Figure_01_02_013.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\"> <strong>Figure 15.<\/strong> For the <strong>absolute value function<\/strong>[latex]\\,f\\left(x\\right)=|x|,\\,[\/latex]there is no restriction on[latex]\\,x.\\,[\/latex]However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.[\/caption]<\/div>\n<div id=\"Figure_01_02_014\" 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\/19134904\/CNX_Precalc_Figure_01_02_014.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\"> <strong>Figure 16. <\/strong>For the <strong>quadratic function<\/strong>[latex]\\,f\\left(x\\right)={x}^{2},\\,[\/latex]the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.[\/caption]<\/div>\n<div id=\"Figure_01_02_015\" 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\/19134918\/CNX_Precalc_Figure_01_02_015.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\"> <strong>Figure 17. <\/strong>For the <strong>cubic function<\/strong>[latex]\\,f\\left(x\\right)={x}^{3},\\,[\/latex]the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.[\/caption]<\/div>\n<div id=\"Figure_01_02_016\" 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\/19134921\/CNX_Precalc_Figure_01_02_016.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\"> <strong>Figure 18. <\/strong>For the <strong>reciprocal function<\/strong>[latex]\\,f\\left(x\\right)=\\frac{1}{x},\\,[\/latex]we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write[latex]\\left\\{x|\\text{ }x\\ne 0\\right\\},[\/latex]the set of all real numbers that are not zero.[\/caption]<\/div>\n<div id=\"Figure_01_02_017\" 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\/19134940\/CNX_Precalc_Figure_01_02_017.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\"> <strong>Figure 19. <\/strong>For the <strong>reciprocal squared function<\/strong>[latex]\\,f\\left(x\\right)=\\frac{1}{{x}^{2}},[\/latex]we cannot divide by [latex]0,[\/latex] so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.[\/caption]<\/div>\n<div id=\"Figure_01_02_018\" 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\/19134942\/CNX_Precalc_Figure_01_02_018.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\"> <strong>Figure 20. <\/strong>For the <strong>square root function<\/strong>[latex]\\,f\\left(x\\right)=\\sqrt[]{x},\\,[\/latex]we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number[latex]\\,x\\,[\/latex]is defined to be positive, even though the square of the negative number[latex]\\,-\\sqrt{x}\\,[\/latex]also gives us[latex]\\,x.[\/latex][\/caption]<\/div>\n<div id=\"Figure_01_02_019\" 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\/19134947\/CNX_Precalc_Figure_01_02_019.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\"> <strong>Figure 21. <\/strong>For the <strong>cube root function<\/strong>[latex]\\,f\\left(x\\right)=\\sqrt[3]{x},\\,[\/latex]the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).[\/caption]<\/div>\n<div id=\"fs-id1165137462732\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137611181\"><strong>Given the formula for a function, determine the domain and range.<\/strong><\/p>\n\n<ol id=\"fs-id1165137405229\" type=\"1\">\n \t<li>Exclude from the domain any input values that result in division by zero.<\/li>\n \t<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\n \t<li>Use the valid input values to determine the range of the output values.<\/li>\n \t<li>Look at the function graph and table values to confirm the actual function behavior.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137558723\">\n<div id=\"fs-id1165137464274\">\n<h3>Finding the Domain and Range Using Toolkit Functions<\/h3>\n<p id=\"fs-id1165135613224\">Find the domain and range of[latex]\\,f\\left(x\\right)=2{x}^{3}-x.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135458670\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135458670\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135458670\"]\n<p id=\"fs-id1165137527861\">There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.<\/p>\n<p id=\"fs-id1165135208585\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]and the range is also[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137448155\">\n<div id=\"fs-id1165137661316\">\n<h3>Finding the Domain and Range<\/h3>\n<p id=\"fs-id1165137419507\">Find the domain and range of[latex]\\,f\\left(x\\right)=\\frac{2}{x+1}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137871182\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137871182\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137871182\"]\n<p id=\"fs-id1165137855321\">We cannot evaluate the function at[latex]\\,-1\\,[\/latex]because division by zero is undefined. The domain is[latex]\\,\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right).\\,[\/latex]Because the function is never zero, we exclude 0 from the range. The range is[latex]\\,\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137574740\">\n<div id=\"fs-id1165135641583\">\n<h3>Finding the Domain and Range<\/h3>\n<p id=\"fs-id1165137661054\">Find the domain and range of[latex]\\,f\\left(x\\right)=2\\sqrt{x+4}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137584342\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137584342\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137584342\"]\n<p id=\"fs-id1165137596350\">We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\n\n<div id=\"eip-id1165137567088\" class=\"unnumbered\">[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/div>\n<p id=\"fs-id1165137465335\">The domain of[latex]\\,f\\left(x\\right)\\,[\/latex]is[latex]\\,\\left[-4,\\infty \\right).[\/latex]<\/p>\n<p id=\"fs-id1165137544393\">We then find the range. We know that[latex]\\,f\\left(-4\\right)=0,\\,[\/latex]and the function value increases as[latex]\\,x\\,[\/latex]increases without any upper limit. We conclude that the range of[latex]\\,f\\,[\/latex]is[latex]\\,\\left[0,\\infty \\right).[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137572635\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137437183\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_020\">(Figure)<\/a> represents the function[latex]\\,f.[\/latex]<\/p>\n\n<div id=\"Figure_01_02_020\" 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\/19134949\/CNX_Precalc_Figure_01_02_020.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\"> <strong>Figure 22.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137430800\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137475544\">\n<p id=\"fs-id1165137475545\">Find the domain and range of[latex]\\,f\\left(x\\right)=-\\sqrt{2-x}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137833252\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137833252\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137833252\"]\n<p id=\"fs-id1165137725047\">domain:[latex]\\,\\left(-\\infty ,2\\right];\\,[\/latex]range:[latex]\\,\\left(-\\infty ,0\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440477\" class=\"bc-section section\">\n<h3>Graphing Piecewise-Defined Functions<\/h3>\n<p id=\"fs-id1165137409262\">Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function[latex]\\,f\\left(x\\right)=|x|.\\,[\/latex]With a domain of all real numbers and a range of values greater than or equal to 0, <span class=\"no-emphasis\">absolute value<\/span> can be defined as the <span class=\"no-emphasis\">magnitude<\/span>, or <span class=\"no-emphasis\">modulus<\/span>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\n<p id=\"fs-id1165137558775\">If we input 0, or a positive value, the output is the same as the input.<\/p>\n\n<div id=\"fs-id1165135194329\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=x\\,\\text{if}\\,x\\ge 0[\/latex]<\/div>\n<p id=\"fs-id1165137529947\">If we input a negative value, the output is the opposite of the input.<\/p>\n\n<div id=\"fs-id1165133112779\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=-x\\,\\text{if}\\,x&lt;0[\/latex]<\/div>\n<p id=\"fs-id1165137863778\">Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\n<p id=\"fs-id1165134042316\">We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income[latex]\\,S\\,[\/latex]would be[latex]\\,0.1S\\,[\/latex]if[latex]\\,S\\le \\text{\\$}10\\text{,}000\\,[\/latex]and[latex]\\,\\text{\\$}1000+0.2\\left(S-\\text{\\$}10\\text{,}000\\right)\\,[\/latex]if[latex]\\,S&gt;\\text{\\$}10\\text{,}000.[\/latex]<\/p>\n\n<div id=\"fs-id1165137531241\" class=\"textbox key-takeaways\">\n<h3>Piecewise Function<\/h3>\n<p id=\"fs-id1165135504970\">A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:<\/p>\n\n<div id=\"fs-id1165137482244\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=\\Bigg\\{\\begin{array}{l}\\text{formula 1 if }x\\text{ is in domain 1}\\\\ \\text{formula 2 if }x\\text{ is in domain 2}\\\\ \\text{formula 3 if }x\\text{ is in domain 3}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137543841\">In piecewise notation, the absolute value function is<\/p>\n\n<div id=\"fs-id1165135190749\" class=\"unnumbered aligncenter\">[latex]|x|=\\bigg\\{\\begin{array}{l}x\\text{ if }x\\ge 0\\\\ -x\\text{ if }x&lt;0\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137768426\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137823161\"><strong>Given a piecewise function, write the formula and identify the domain for each interval.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165135443772\" type=\"1\">\n \t<li>Identify the intervals for which different rules apply.<\/li>\n \t<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\n \t<li>Use braces and if-statements to write the function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137452506\">\n<div id=\"fs-id1165135321994\">\n<h3>Writing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137834905\">A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <span class=\"no-emphasis\">function<\/span> relating the number of people,[latex]\\,n,\\,[\/latex]to the cost,[latex]\\,C.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137807421\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137807421\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137807421\"]\n<p id=\"fs-id1165135331729\">Two different formulas will be needed. For <em>n<\/em>-values under 10,[latex]\\,C=5n.\\,[\/latex]For values of[latex]\\,n\\,[\/latex]that are 10 or greater,[latex]\\,C=50.[\/latex]<\/p>\n\n<div id=\"fs-id1165135208951\" class=\"unnumbered aligncenter\">[latex]C\\left(n\\right)=\\left\\{\\begin{array}{ccc}5n&amp; \\text{if}&amp; 0&lt;n&lt;10\\\\ 50&amp; \\text{if}&amp; n\\ge 10\\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135436578\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135196985\">The function is represented in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_021\">(Figure)<\/a>. The graph is a diagonal line from[latex]\\,n=0\\,[\/latex]to[latex]\\,n=10\\,[\/latex]and a constant after that. In this example, the two formulas agree at the meeting point where[latex]\\,n=10,\\,[\/latex]but not all piecewise functions have this property.<\/p>\n\n<div id=\"Figure_01_02_021\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"360\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19134959\/CNX_Precalc_Figure_01_02_021.jpg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\"> <strong>Figure 23.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135436662\">\n<div id=\"fs-id1165135436664\">\n<h3>Working with a Piecewise Function<\/h3>\n<p id=\"fs-id1165137938645\">A cell phone company uses the function below to determine the cost,[latex]\\,C,\\,[\/latex]in dollars for[latex]\\,g\\,[\/latex]gigabytes of data transfer.<\/p>\n\n<div id=\"fs-id1165137660470\" class=\"unnumbered aligncenter\">[latex]C\\left(g\\right)=\\left\\{\\begin{array}{ccc}25&amp; \\text{if}&amp; 0&lt;g&lt;2\\\\ 25+10\\left(g-2\\right)&amp; \\text{if}&amp; g\\ge 2\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135193798\">Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135177567\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135177567\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135177567\"]\n<p id=\"fs-id1165134373545\">To find the cost of using 1.5 gigabytes of data,[latex]\\,C\\left(1.5\\right),\\,[\/latex]we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.<\/p>\n\n<div id=\"fs-id1165134300204\" class=\"unnumbered aligncenter\">[latex]C\\left(1.5\\right)=\\text{\\$}25[\/latex]<\/div>\n<p id=\"fs-id1165135440213\">To find the cost of using 4 gigabytes of data,[latex]\\,C\\left(4\\right),\\,[\/latex]we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\n\n<div id=\"fs-id1165135383665\" class=\"unnumbered aligncenter\">[latex]C\\left(4\\right)=25+10\\left(4-2\\right)=\\text{\\$}45[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137634432\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137601265\">The function is represented in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_022\">(Figure)<\/a>. We can see where the function changes from a constant to a shifted and stretched identity at[latex]\\,g=2.\\,[\/latex]We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.<\/p>\n\n<div id=\"Figure_01_02_022\" 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\/19135002\/CNX_Precalc_Figure_01_02_022.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\"> <strong>Figure 24.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600493\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135532516\"><strong>Given a piecewise function, sketch a graph.<\/strong><\/p>\n\n<ol id=\"fs-id1165137588539\" type=\"1\">\n \t<li>Indicate on the <em>x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\n \t<li>For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137781618\">\n<div id=\"fs-id1165135412870\">\n<h3>Graphing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137838785\">Sketch a graph of the function.<\/p>\n\n<div id=\"fs-id1165137475346\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}{x}^{2}&amp; \\text{if}&amp; x\\le 1\\\\ 3&amp; \\text{if}&amp; 1&lt;x\\le 2\\\\ x&amp; \\text{if}&amp; x&gt;2\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135487148\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135487148\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135487148\"]\n<p id=\"fs-id1165135487150\">Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.<\/p>\n<p id=\"fs-id1165137642848\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_023\">(Figure)<\/a> shows the three components of the piecewise function graphed on separate coordinate systems.<\/p>\n\n<div id=\"Figure_01_02_023\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"974\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135010\/CNX_Precalc_Figure_01_02_023abc.jpg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\"> <strong>Figure 25. <\/strong>(a)[latex]\\,f\\left(x\\right)={x}^{2}\\text{ if }x\\le 1;\\,[\/latex](b)[latex]\\,f\\left(x\\right)=3\\text{ if 1&lt; }x\\le 2;\\,[\/latex](c)[latex]\\,f\\left(x\\right)=x\\text{ if }x&gt;2[\/latex][\/caption]<\/div>\n<p id=\"fs-id1165137676209\">Now that we have sketched each piece individually, we combine them in the same coordinate plane. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_026\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_02_026\" 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\/19135018\/CNX_Precalc_Figure_01_02_026.jpg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\"> <strong>Figure 26.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188517\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134389893\">Note that the graph does pass the vertical line test even at[latex]\\,x=1\\,[\/latex]and[latex]\\,x=2\\,[\/latex]because the points [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(2,2\\right)[\/latex] are not part of the graph of the function, though [latex]\\left(1,1\\right)[\/latex]\nand [latex]\\left(2,\\,3\\right)[\/latex] are.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762558\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137692562\">\n<p id=\"fs-id1165137692563\">Graph the following piecewise function.<\/p>\n\n<div id=\"fs-id1165137433350\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}{x}^{3}&amp; \\text{if}&amp; x&lt;-1\\\\ -2&amp; \\text{if}&amp; -1&lt;x&lt;4\\\\ \\sqrt{x}&amp; \\text{if}&amp; x&gt;4\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137784656\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137784656\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137784656\"]<span id=\"fs-id1165134302462\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135020\/CNX_Precalc_Figure_01_02_027.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810682\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137527804\"><strong>Can more than one formula from a piecewise function be applied to a value in the domain?<\/strong><\/p>\n<p id=\"fs-id1165137464467\"><em>No. Each value corresponds to one equation in a piecewise formula.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165135190393\" class=\"precalculus media\">\n<p id=\"fs-id1165137627040\">Access these online resources for additional instruction and practice with domain and range.<\/p>\n\n<ul id=\"fs-id1165135189954\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/domainsqroot\">Domain and Range of Square Root Functions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/determinedomain\">Determining Domain and Range<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/drgraph\">Find Domain and Range Given the Graph<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/drtable\">Find Domain and Range Given a Table<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/drcoordinate\">Find Domain and Range Given Points on a Coordinate Plane<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134077347\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137591772\">\n \t<li>The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.<\/li>\n \t<li>The domain of a function can be determined by listing the input values of a set of ordered pairs. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_01\">(Figure)<\/a>.<\/li>\n \t<li>The domain of a function can also be determined by identifying the input values of a function written as an equation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_02\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_03\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_04\">(Figure)<\/a>.<\/li>\n \t<li>Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_05\">(Figure)<\/a>.<\/li>\n \t<li>For many functions, the domain and range can be determined from a graph. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_07\">(Figure)<\/a>.<\/li>\n \t<li>An understanding of toolkit functions can be used to find the domain and range of related functions. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_08\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_09\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_10\">(Figure)<\/a>.<\/li>\n \t<li>A piecewise function is described by more than one formula. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_11\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_12\">(Figure)<\/a>.<\/li>\n \t<li>A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_13\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135176628\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135172218\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137665109\">\n<div id=\"fs-id1165135245908\">\n<p id=\"fs-id1165135245910\">Why does the domain differ for different functions?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134199600\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134199600\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134199600\"]\n<p id=\"fs-id1165135613709\">The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440209\">\n<div id=\"fs-id1165135533141\">\n<p id=\"fs-id1165135533143\">How do we determine the domain of a function defined by an equation?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635386\">\n<div id=\"fs-id1165135390940\">\n<p id=\"fs-id1165135390942\">Explain why the domain of[latex]\\,f\\left(x\\right)=\\sqrt[3]{x}\\,[\/latex]is different from the domain of[latex]\\,f\\left(x\\right)=\\sqrt[]{x}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137727146\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137727146\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137727146\"]\n<p id=\"fs-id1165137727148\">There is no restriction on[latex]\\,x\\,[\/latex]for[latex]\\,f\\left(x\\right)=\\sqrt[3]{x}\\,[\/latex]because you can take the cube root of any real number. So the domain is all real numbers,[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]When dealing with the set of real numbers, you cannot take the square root of negative numbers. So[latex]\\,x[\/latex]-values are restricted for[latex]\\,f\\left(x\\right)=\\sqrt[]{x}\\,[\/latex]to nonnegative numbers and the domain is[latex]\\,\\left[0,\\infty \\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042454\">\n<div id=\"fs-id1165134042457\">\n<p id=\"fs-id1165137438149\">When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134211324\">\n<div id=\"fs-id1165137446310\">\n<p id=\"fs-id1165137446313\">How do you graph a piecewise function?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137574335\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137574335\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137574335\"]\n<p id=\"fs-id1165135415726\">Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the[latex]\\,x[\/latex]-axis and[latex]\\,y[\/latex]-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate[latex]\\,-\\infty \\,[\/latex]or[latex]\\,\\text{ }\\infty .\\,[\/latex]Combine the graphs to find the graph of the piecewise function.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771069\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137408926\">For the following exercises, find the domain of each function using interval notation.<\/p>\n\n<div id=\"fs-id1165137833819\">\n<div id=\"fs-id1165137833821\">\n<p id=\"fs-id1165135500745\">[latex]f\\left(x\\right)=-2x\\left(x-1\\right)\\left(x-2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137854912\">\n<div id=\"fs-id1165134312130\">\n<p id=\"fs-id1165134312132\">[latex]f\\left(x\\right)=5-2{x}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137731586\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137731586\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137731586\"]\n<p id=\"fs-id1165137731589\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135512534\">\n<div id=\"fs-id1165135512537\">\n<p id=\"fs-id1165137804475\">[latex]f\\left(x\\right)=3\\sqrt{x-2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473385\">\n<div id=\"fs-id1165137473388\">\n<p id=\"fs-id1165134374059\">[latex]f\\left(x\\right)=3-\\sqrt{6-2x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137451053\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137451053\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137451053\"]\n<p id=\"fs-id1165137451055\">[latex]\\left(-\\infty ,3\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192268\">\n<div id=\"fs-id1165135192270\">\n<p id=\"fs-id1165137725224\">[latex]f\\left(x\\right)=\\sqrt{4-3x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137629066\">\n<div id=\"fs-id1165137483196\">\n<p id=\"fs-id1165137483198\">[latex]\\begin{array}{l}\\\\ f\\left(x\\right)=\\sqrt[]{{x}^{2}+4}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134192936\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134192936\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134192936\"]\n<p id=\"fs-id1165137463777\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137807107\">\n<div id=\"fs-id1165137551129\">\n<p id=\"fs-id1165137551131\">[latex]f\\left(x\\right)=\\sqrt[3]{1-2x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134259277\">\n<div id=\"fs-id1165134259279\">\n<p id=\"fs-id1165135186001\">[latex]f\\left(x\\right)=\\sqrt[3]{x-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134058389\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134058389\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134058389\"]\n<p id=\"fs-id1165137836596\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503751\">\n<div id=\"fs-id1165134156030\">\n<p id=\"fs-id1165134156032\">[latex]f\\left(x\\right)=\\frac{9}{x-6}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133276237\">\n<div id=\"fs-id1165133276240\">\n<p id=\"fs-id1165137784864\">[latex]f\\left(x\\right)=\\frac{3x+1}{4x+2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137454548\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137454548\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137454548\"]\n<p id=\"fs-id1165134170171\">[latex]\\left(-\\infty ,-\\frac{1}{2}\\right)\\cup \\left(-\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810520\">\n<div id=\"fs-id1165137810522\">\n<p id=\"fs-id1165137532795\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x+4}}{x-4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135548992\">\n<div id=\"fs-id1165135634123\">\n<p id=\"fs-id1165135634125\">[latex]f\\left(x\\right)=\\frac{x-3}{{x}^{2}+9x-22}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137528909\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137528909\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137528909\"]\n<p id=\"fs-id1165137528911\">[latex]\\left(-\\infty ,-11\\right)\\cup \\left(-11,2\\right)\\cup \\left(2,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593402\">\n<div id=\"fs-id1165135593404\">\n<p id=\"fs-id1165137771850\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-x-6}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191342\">\n<div id=\"fs-id1165134284474\">\n<p id=\"fs-id1165134284476\">[latex]f\\left(x\\right)=\\frac{2{x}^{3}-250}{{x}^{2}-2x-15}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135256053\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135256053\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135256053\"]\n<p id=\"fs-id1165135256055\">[latex]\\left(-\\infty ,-3\\right)\\cup \\left(-3,5\\right)\\cup \\left(5,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137921795\">\n<div id=\"fs-id1165137921797\">\n<p id=\"fs-id1165137532172\">[latex]\\frac{5}{\\sqrt{x-3}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476914\">\n<div id=\"fs-id1165137476916\">\n<p id=\"fs-id1165137726504\">[latex]\\frac{2x+1}{\\sqrt{5-x}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137647829\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137647829\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137647829\"]\n<p id=\"fs-id1165137564959\">[latex]\\left(-\\infty ,5\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135185292\">\n<div id=\"fs-id1165137640755\">\n<p id=\"fs-id1165137640757\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x-4}}{\\sqrt{x-6}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135252252\">\n<div id=\"fs-id1165137611840\">\n<p id=\"fs-id1165137611842\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137611238\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137611238\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137611238\"]\n<p id=\"fs-id1165137538970\">[latex]\\left[6,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601712\">\n<div id=\"fs-id1165137601714\">\n<p id=\"fs-id1165137657487\">[latex]f\\left(x\\right)=\\frac{x}{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628472\">\n<div id=\"fs-id1165137651574\">\n<p id=\"fs-id1165137651576\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-9x}{{x}^{2}-81}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135188135\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135188135\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135188135\"]\n<p id=\"fs-id1165137809882\">[latex]\\left(-\\infty ,-9\\right)\\cup \\left(-9,9\\right)\\cup \\left(9,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137469452\">\n<div id=\"fs-id1165137469454\">\n<p id=\"fs-id1165137635293\">Find the domain of the function[latex]\\,f\\left(x\\right)=\\sqrt{2{x}^{3}-50x}\\,[\/latex]by:<\/p>\n\n<ol id=\"fs-id1165137938832\" type=\"a\">\n \t<li>using algebra.<\/li>\n \t<li>graphing the function in the radicand and determining intervals on the <em>x<\/em>-axis for which the radicand is nonnegative.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137580833\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135186809\">For the following exercises, write the domain and range of each function using interval notation.<\/p>\n\n<div id=\"fs-id1165135168172\">\n<div id=\"fs-id1165137647479\"><span id=\"fs-id1165137891294\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135032\/CNX_Precalc_Figure_01_02_202.jpg\" alt=\"Graph of a function from (2, 8].\"><\/span><\/div>\n<div id=\"fs-id1165137820038\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137820038\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137820038\"]\n<p id=\"fs-id1165137424631\">domain:[latex]\\,\\left(2,8\\right],\\,[\/latex]range[latex]\\,\\left[6,8\\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160181\">\n<div id=\"fs-id1165135160183\"><span id=\"fs-id1165137837830\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135034\/CNX_Precalc_Figure_01_02_203.jpg\" alt=\"Graph of a function from [4, 8).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137723404\">\n<div id=\"fs-id1165137809982\"><span id=\"fs-id1165137733767\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135035\/CNX_Precalc_Figure_01_02_204.jpg\" alt=\"Graph of a function from [-4, 4].\"><\/span><\/div>\n<div id=\"fs-id1165137847285\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137847285\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137847285\"]\n<p id=\"fs-id1165137541038\">domain:[latex]\\,\\left[-4,\\text{ 4],}\\,[\/latex]range:[latex]\\,\\left[0,\\text{ 2]}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137590678\">\n<div id=\"fs-id1165134168421\"><span id=\"fs-id1165137837060\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135045\/CNX_Precalc_Figure_01_02_205.jpg\" alt=\"Graph of a function from [2, 6].\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137737326\">\n<div id=\"fs-id1165137737328\"><span id=\"fs-id1165134129572\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135058\/CNX_Precalc_Figure_01_02_206.jpg\" alt=\"Graph of a function from [-5, 3).\"><\/span><\/div>\n<div id=\"fs-id1165137657479\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137657479\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137657479\"]\n<p id=\"fs-id1165137657482\">domain:[latex]\\,\\left[-5,\\text{ }3\\right),\\,[\/latex]range:[latex]\\,\\left[0,2\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404973\">\n<div id=\"fs-id1165137404975\"><span id=\"fs-id1165134305418\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135100\/CNX_Precalc_Figure_01_02_207.jpg\" alt=\"Graph of a function from [-3, 2).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137544188\">\n<div id=\"fs-id1165137437269\"><span id=\"fs-id1165137447903\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135103\/CNX_Precalc_Figure_01_02_208.jpg\" alt=\"Graph of a function from (-infinity, 2].\"><\/span><\/div>\n<div id=\"fs-id1165137445711\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137445711\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137445711\"]\n<p id=\"fs-id1165137445713\">domain:[latex]\\,\\left(-\\infty ,1\\right],\\,[\/latex]range:[latex]\\,\\left[0,\\infty \\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176309\">\n<div id=\"fs-id1165134323791\"><span id=\"fs-id1165135192955\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135105\/CNX_Precalc_Figure_01_02_209.jpg\" alt=\"Graph of a function from [-4, infinity).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137642580\">\n<div id=\"fs-id1165137642582\"><span id=\"fs-id1165134482733\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135116\/CNX_Precalc_Figure_01_02_210.jpg\" alt=\"Graph of a function from [-6, -1\/6]U[1\/6, 6]\/.\"><\/span><\/div>\n<div id=\"fs-id1165134043582\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134043582\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134043582\"]\n<p id=\"fs-id1165135335983\">domain:[latex]\\,\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right];\\,[\/latex]range:[latex]\\,\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right]\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137442385\">\n<div id=\"fs-id1165137812572\"><span id=\"fs-id1165137645308\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135118\/CNX_Precalc_Figure_01_02_211.jpg\" alt=\"Graph of a function from (-2.5, infinity).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137851981\">\n<div id=\"fs-id1165137851983\"><span id=\"fs-id1165137602824\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135121\/CNX_Precalc_Figure_01_02_212.jpg\" alt=\"Graph of a function from [-3, infinity).\"><\/span><\/div>\n<div id=\"fs-id1165137575572\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137575572\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137575572\"]\n<p id=\"fs-id1165137601170\">domain:[latex]\\,\\left[-3,\\text{ }\\infty \\right);\\,[\/latex]range:[latex]\\,\\left[0,\\infty \\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137785119\">For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\n\n<div id=\"fs-id1165137462167\">\n<div id=\"fs-id1165137408525\">\n<p id=\"fs-id1165137408527\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}x+1\\hfill &amp; \\text{if}\\hfill &amp; x&lt;-2\\hfill \\\\ -2x-3\\hfill &amp; \\text{if}\\hfill &amp; x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137562309\">\n<div id=\"fs-id1165134328320\">\n<p id=\"fs-id1165134328322\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}2x-1\\hfill &amp; \\text{if}\\hfill &amp; x&lt;1\\hfill \\\\ 1+x\\hfill &amp; \\text{if}\\hfill &amp; x\\ge 1\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135481131\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135481131\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135481131\"]\n<p id=\"fs-id1165135481133\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<span id=\"fs-id1165137662700\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135123\/CNX_Precalc_Figure_01_02_214.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628033\">\n<div id=\"fs-id1165137658060\">\n<p id=\"fs-id1165137658062\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}x+1\\,\\,\\text{if}\\,\\,x&lt;0\\\\ x-1\\,\\,\\text{if}\\,\\,\\,x&gt;0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135641679\">\n<div id=\"fs-id1165135641681\">\n<p id=\"fs-id1165133402089\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}3&amp; \\text{if}&amp; x&lt;0\\\\ \\sqrt{x}&amp; \\text{if}&amp; x\\ge 0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137500956\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137500956\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137500956\"]\n<p id=\"fs-id1165135532432\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<span id=\"fs-id1165137474386\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135125\/CNX_Precalc_Figure_01_02_216.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192719\">\n<div id=\"fs-id1165135192721\">\n<p id=\"fs-id1165137400953\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}{x}^{2}\\text{ if }x&lt;0\\\\ 1-x\\text{ if }x&gt;0\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137594981\">\n<div id=\"fs-id1165135210029\">\n<p id=\"fs-id1165135210031\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{r}\\hfill \\begin{array}{r}\\hfill {x}^{2}\\\\ \\hfill x+2\\end{array}\\end{array}\\,\\,\\begin{array}{l}\\text{if}\\,\\,\\,\\,\\,x&lt;0\\hfill \\\\ \\text{if}\\,\\,\\,\\,\\,x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137667233\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137667233\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137667233\"]\n<p id=\"fs-id1165135382142\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<span id=\"fs-id1165135188662\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135134\/CNX_Precalc_Figure_01_02_218.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137571389\">\n<div id=\"fs-id1165137433000\">\n<p id=\"fs-id1165137433002\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}x+1&amp; \\text{if}&amp; x&lt;1\\\\ {x}^{3}&amp; \\text{if}&amp; x\\ge 1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407891\">\n<div id=\"fs-id1165137554125\">\n<p id=\"fs-id1165137554127\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}|x|\\\\ 1\\end{array}\\begin{array}{l}\\,\\,\\,\\text{if}\\,\\,\\,x&lt;2\\hfill \\\\ \\,\\,\\,\\text{if}\\,\\,\\,x\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137401041\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137401041\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137401041\"]\n<p id=\"fs-id1165134252896\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<span id=\"fs-id1165135432997\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135140\/CNX_Precalc_Figure_01_02_220.jpg\" alt=\"Graph of f(x).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134118450\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165135188383\">For the following exercises, given each function [latex]f,[\/latex]evaluate [latex]f\\left(-3\\right),\\,f\\left(-2\\right),\\,f\\left(-1\\right),[\/latex] and [latex]f\\left(0\\right).[\/latex]<\/p>\n\n<div id=\"fs-id1165137471865\">\n<div id=\"fs-id1165137471867\">\n<p id=\"fs-id1165134043731\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}x+1\\hfill &amp; \\text{if}\\hfill &amp; x&lt;-2\\hfill \\\\ -2x-3\\hfill &amp; \\text{if}\\hfill &amp; x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122954\">\n<div id=\"fs-id1165134122956\">\n<p id=\"fs-id1165135168423\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{cc}1&amp; \\text{if }x\\le -3\\\\ 0&amp; \\text{if }x&gt;-3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137804494\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137804494\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137804494\"]\n<p id=\"fs-id1165137804496\">[latex]\\begin{array}{cccc}f\\left(-3\\right)=1;&amp; f\\left(-2\\right)=0;&amp; f\\left(-1\\right)=0;&amp; f\\left(0\\right)=0\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556768\">\n<div id=\"fs-id1165137423742\">\n<p id=\"fs-id1165137423744\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{cc}-2{x}^{2}+3&amp; \\text{if }x\\le -1\\\\ 5x-7&amp; \\text{if }x&gt;-1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137469026\">For the following exercises, given each function[latex]\\,f,\\,[\/latex]evaluate[latex]f\\left(-1\\right),\\,f\\left(0\\right),\\,f\\left(2\\right),\\,[\/latex]and[latex]\\,f\\left(4\\right).[\/latex]<\/p>\n\n<div>\n<div id=\"fs-id1165134380353\">\n<p id=\"fs-id1165137678245\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}7x+3\\hfill &amp; \\text{if}\\hfill &amp; x&lt;0\\hfill \\\\ 7x+6\\hfill &amp; \\text{if}\\hfill &amp; x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137476514\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137476514\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137476514\"]\n<p id=\"fs-id1165137476516\">[latex]\\begin{array}{cccc}f\\left(-1\\right)=-4;&amp; f\\left(0\\right)=6;&amp; f\\left(2\\right)=20;&amp; f\\left(4\\right)=34\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137693713\">\n<div id=\"fs-id1165137679373\">\n<p id=\"fs-id1165137679375\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}{x}^{2}-2&amp; \\text{if}&amp; x&lt;2\\\\ 4+|x-5|&amp; \\text{if}&amp; x\\ge 2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137715004\">\n<div id=\"fs-id1165137715006\">\n<p id=\"fs-id1165137715008\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}5x&amp; \\text{if}&amp; x&lt;0\\\\ 3&amp; \\text{if}&amp; 0\\le x\\le 3\\\\ {x}^{2}&amp; \\text{if}&amp; x&gt;3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135699157\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135699157\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135699157\"]\n<p id=\"fs-id1165137401550\">[latex]\\begin{array}{cccc}f\\left(-1\\right)=-5;&amp; f\\left(0\\right)=3;&amp; f\\left(2\\right)=3;&amp; f\\left(4\\right)=16\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137837869\">For the following exercises, write the domain for the piecewise function in interval notation.<\/p>\n\n<div id=\"fs-id1165137837872\">\n<div id=\"fs-id1165135341427\">\n<p id=\"fs-id1165135341429\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}x+1\\,\\,\\,\\,\\,\\text{ if}\\,\\,x&lt;-2\\\\ -2x-3\\,\\,\\text{if}\\,\\,x\\ge -2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704661\">\n<div id=\"fs-id1165137704664\">\n<p id=\"fs-id1165137704666\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}{x}^{2}-2\\,\\,\\,\\,\\,\\text{ if}\\,\\,x&lt;1\\\\ -{x}^{2}+2\\,\\,\\text{if}\\,\\,x&gt;1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135420410\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135420410\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135420410\"]\n<p id=\"fs-id1165135570357\">domain:[latex]\\,\\left(-\\infty ,1\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137772429\">\n<div id=\"fs-id1165137772431\">\n<p id=\"fs-id1165137675983\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}2x-3\\\\ -3{x}^{2}\\end{array}\\,\\,\\begin{array}{c}\\text{if}\\,\\,\\,x&lt;0\\\\ \\text{if}\\,\\,\\,x\\ge 2\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194497\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165137780865\">\n<div id=\"fs-id1165137780867\">\n<p id=\"fs-id1165135641711\">Graph[latex]\\,y=\\frac{1}{{x}^{2}}\\,[\/latex]on the viewing window[latex]\\,\\left[-0.5,-0.1\\right]\\,[\/latex]and[latex]\\,\\left[0.1,0.5\\right].\\,[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137501974\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"749724\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"749724\"]<img src=\"https:\/\/cnx.org\/resources\/e9ff86fef34aa7f3335b023686c29d96c31810c9\/CNX_Precalc_Figure_01_02_221.jpg\" alt=\"Graph of the equation from [-0.5, -0.1].\">\n<p id=\"fs-id1165135191028\">window:<span id=\"MathJax-Element-2804-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51126\" class=\"math\"><span id=\"MathJax-Span-51127\" class=\"mrow\"><span id=\"MathJax-Span-51128\" class=\"semantics\"><span id=\"MathJax-Span-51129\" class=\"mrow\"><span id=\"MathJax-Span-51130\" class=\"mrow\"><span id=\"MathJax-Span-51131\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51132\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51133\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-51134\" class=\"mn\">0.5<\/span><span id=\"MathJax-Span-51135\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51136\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-51137\" class=\"mn\">0.1<\/span><span id=\"MathJax-Span-51138\" class=\"mo\">]<\/span><span id=\"MathJax-Span-51139\" class=\"mo\">;<\/span><span id=\"MathJax-Span-51140\" class=\"mtext\">\u2009<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[\u22120.5,\u22120.1];\u2009\u2009[\u22120.5,\u22120.1];\u2009<\/span><\/span> range:<span id=\"MathJax-Element-2805-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51141\" class=\"math\"><span id=\"MathJax-Span-51142\" class=\"mrow\"><span id=\"MathJax-Span-51143\" class=\"semantics\"><span id=\"MathJax-Span-51144\" class=\"mrow\"><span id=\"MathJax-Span-51145\" class=\"mrow\"><span id=\"MathJax-Span-51146\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51147\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51148\" class=\"mn\">4<\/span><span id=\"MathJax-Span-51149\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51150\" class=\"mtext\">&nbsp;<\/span><span id=\"MathJax-Span-51151\" class=\"mn\">100<\/span><span id=\"MathJax-Span-51152\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[4,&nbsp;100]\u2009[4,&nbsp;100]<\/span><\/span><\/p>\n<span id=\"fs-id1165137442910\"><img src=\"https:\/\/cnx.org\/resources\/1958a0a7420a22e38d7733e4c3481051536b1e52\/CNX_Precalc_Figure_01_02_222.jpg\" alt=\"Graph of the equation from [0.1, 0.5].\"> <\/span>\n<p id=\"fs-id1165134378637\">window:<span id=\"MathJax-Element-2806-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51153\" class=\"math\"><span id=\"MathJax-Span-51154\" class=\"mrow\"><span id=\"MathJax-Span-51155\" class=\"semantics\"><span id=\"MathJax-Span-51156\" class=\"mrow\"><span id=\"MathJax-Span-51157\" class=\"mrow\"><span id=\"MathJax-Span-51158\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51159\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51160\" class=\"mn\">0.1<\/span><span id=\"MathJax-Span-51161\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51162\" class=\"mtext\">&nbsp;<\/span><span id=\"MathJax-Span-51163\" class=\"mn\">0.5<\/span><span id=\"MathJax-Span-51164\" class=\"mo\">]<\/span><span id=\"MathJax-Span-51165\" class=\"mo\">;<\/span><span id=\"MathJax-Span-51166\" class=\"mtext\">\u2009<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[0.1,&nbsp;0.5];\u2009\u2009[0.1,&nbsp;0.5];\u2009<\/span><\/span> range:<span id=\"MathJax-Element-2807-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51167\" class=\"math\"><span id=\"MathJax-Span-51168\" class=\"mrow\"><span id=\"MathJax-Span-51169\" class=\"semantics\"><span id=\"MathJax-Span-51170\" class=\"mrow\"><span id=\"MathJax-Span-51171\" class=\"mrow\"><span id=\"MathJax-Span-51172\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51173\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51174\" class=\"mn\">4<\/span><span id=\"MathJax-Span-51175\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51176\" class=\"mtext\">&nbsp;<\/span><span id=\"MathJax-Span-51177\" class=\"mn\">100<\/span><span id=\"MathJax-Span-51178\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[4,&nbsp;100]\u2009[4,&nbsp;100]<\/span><\/span><\/p>\n<p id=\"fs-id1165134378637\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131911953\">\n<div id=\"fs-id1165137842479\">\n<p id=\"fs-id1165137842481\">Graph[latex]\\,y=\\frac{1}{x}\\,[\/latex]on the viewing window[latex]\\,\\left[-0.5,-0.1\\right]\\,[\/latex]and[latex]\\,\\left[0.1,\\text{ }0.5\\right].\\,[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137733672\" class=\"bc-section section\">\n<h4>Extension<\/h4>\n<div id=\"fs-id1165137442197\">\n<div id=\"fs-id1165133221851\">\n<p id=\"fs-id1165133221853\">Suppose the range of a function[latex]\\,f\\,[\/latex]is[latex]\\,\\left[-5,\\text{ }8\\right].\\,[\/latex]What is the range of[latex]\\,|f\\left(x\\right)|?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134555582\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134555582\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134555582\"]\n<p id=\"fs-id1165134555584\">[latex]\\left[0,\\text{ }8\\right][\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137679047\">\n<div id=\"fs-id1165137679049\">\n<p id=\"fs-id1165133410011\">Create a function in which the range is all nonnegative real numbers.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209378\">\n<div id=\"fs-id1165135209380\">\n<p id=\"fs-id1165137645593\">Create a function in which the domain is[latex]\\,x&gt;2.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133210812\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133210812\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133210812\"]\n<p id=\"fs-id1165137779064\">Many answers. One function is[latex]\\,f\\left(x\\right)=\\frac{1}{\\sqrt{x-2}}.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832031\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135511303\">\n<div id=\"fs-id1165135511305\">\n<p id=\"fs-id1165135336103\">The height[latex]\\,h\\,[\/latex]of a projectile is a function of the time[latex]\\,t\\,[\/latex]it is in the air. The height in feet for[latex]\\,t\\,[\/latex]seconds is given by the function[latex]h\\left(t\\right)=-16{t}^{2}+96t.[\/latex]\nWhat is the domain of the function? What does the domain mean in the context of the problem?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137446701\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137446701\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137446701\"]\n<p id=\"fs-id1165137758760\">The domain is[latex]\\,\\left[0,\\text{ }6\\right];\\,[\/latex]it takes 6 seconds for the projectile to leave the ground and return to the ground<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137406705\">\n<div id=\"fs-id1165137406708\">\n<p id=\"fs-id1165133045371\">The cost in dollars of making[latex]\\,x\\,[\/latex]items is given by the function[latex]\\,C\\left(x\\right)=10x+500.[\/latex]<\/p>\n\n<ol id=\"fs-id1165137862357\" type=\"a\">\n \t<li>The fixed cost is determined when zero items are produced. Find the fixed cost for this item.<\/li>\n \t<li>What is the cost of making 25 items?<\/li>\n \t<li>Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function,[latex]\\,C\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl>\n \t<dt>interval notation<\/dt>\n \t<dd id=\"fs-id1165135190252\">a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135487256\">\n \t<dt>piecewise function<\/dt>\n \t<dd id=\"fs-id1165137452169\">a function in which more than one formula is used to define the output<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137863188\">\n \t<dt>set-builder notation<\/dt>\n \t<dd id=\"fs-id1165137863193\">a method of describing a set by a rule that all of its members obey; it takes the form[latex]\\,\\left\\{x|\\,\\text{statement about }x\\right\\}[\/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>Find the domain of a function defined by an equation.<\/li>\n<li>Graph piecewise-defined functions.<\/li>\n<\/ul>\n<\/div>\n<p>If you\u2019re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time\u2014<em>I am Legend<\/em>, <em>Hannibal<\/em>, <em>The Ring<\/em>, <em>The Grudge<\/em>, and <em>The Conjuring<\/em>. <a class=\"autogenerated-content\" href=\"#Figure_01_02_001\">(Figure)<\/a> shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <span class=\"no-emphasis\">domain<\/span> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.<\/p>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/f29850a56c5a5ec5355fe59ed2e7ce1a3bb85627\/CNX_Precalc_Figure_01_02_001.jpg\" alt=\"Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year\" width=\"975\" height=\"402\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Based on data compiled by www.the-numbers.com.<\/figcaption><\/figure>\n<p><a class=\"footnote\" title=\"The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014\" id=\"return-footnote-55-1\" href=\"#footnote-55-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<div id=\"fs-id1165135193832\" class=\"bc-section section\">\n<h3>Finding the Domain of a Function Defined by an Equation<\/h3>\n<p id=\"fs-id1165135445896\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/55f2e8ec-a982-4586-9d48-a2f43d7b4107\">Functions and Function Notation<\/a>, we were introduced to the concepts of <span class=\"no-emphasis\">domain and range<\/span>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \u201cholding area\u201d that contains \u201craw materials\u201d for a \u201cfunction machine\u201d and the range as another \u201cholding area\u201d for the machine\u2019s products. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_002\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_02_002\" 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\/19134739\/CNX_Precalc_Figure_01_02_002.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137761714\">We can write the <span class=\"no-emphasis\">domain and range<\/span> in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write[latex]\\,\\left(0,\\text{ }100\\right].\\,[\/latex]We will discuss interval notation in greater detail later.<\/p>\n<p id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\n<ul id=\"fs-id1165135673417\">\n<li>The smallest number from the interval is written first.<\/li>\n<li>The largest number in the interval is written second, following a comma.<\/li>\n<li>Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.<\/li>\n<li>Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.<\/li>\n<\/ul>\n<p id=\"fs-id1165137807384\">See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_029\">(Figure)<\/a> for a summary of interval notation.<\/p>\n<div id=\"Figure_01_02_029\" 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\/19134741\/CNX_Precalc_Figure_01_02_029n.jpg\" alt=\"Summary of interval notation.\" width=\"975\" height=\"905\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137661548\">\n<div id=\"fs-id1165137772018\">\n<h3>Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\n<p id=\"fs-id1165137920768\">Find the <span class=\"no-emphasis\">domain<\/span> of the following function:[latex]\\,\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135329797\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135508343\">First identify the input values. The input value is the first coordinate in an <span class=\"no-emphasis\">ordered pair<\/span>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\n<div id=\"fs-id1165137451888\" class=\"unnumbered aligncenter\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137569901\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165135333722\">\n<div id=\"fs-id1165137852040\">\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\n<p id=\"fs-id1165137466017\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137501477\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137704712\">[latex]\\left\\{-5,\\,0,\\,5,\\,10,\\,15\\right\\}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134225655\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134355557\"><strong>Given a function written in equation form, find the domain.<\/strong><\/p>\n<ol id=\"fs-id1165134187286\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\n<li>Write the domain in interval form, if possible.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137767649\">\n<div id=\"fs-id1165137761307\">\n<h3>Finding the Domain of a Function<\/h3>\n<p id=\"fs-id1165137645656\">Find the domain of the function[latex]\\,f\\left(x\\right)={x}^{2}-1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135684349\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137594433\">The input value, shown by the variable[latex]\\,x\\,[\/latex]in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.<\/p>\n<p id=\"fs-id1165135309759\">In interval form, the domain of[latex]\\,f\\,[\/latex]is[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135639906\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137733850\">\n<div id=\"fs-id1165137871971\">\n<p id=\"fs-id1165137871972\">Find the domain of the function:[latex]\\,f\\left(x\\right)=5-x+{x}^{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137809848\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137809849\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417188\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137473617\"><strong>Given a function written in an equation form that includes a fraction, find the domain.<\/strong><\/p>\n<ol id=\"fs-id1165137463251\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for[latex]\\,x\\,[\/latex]. If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\n<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137722406\">\n<div id=\"fs-id1165135484119\">\n<h3>Finding the Domain of a Function Involving a Denominator<\/h3>\n<p id=\"fs-id1165137647592\">Find the <span class=\"no-emphasis\">domain<\/span> of the function[latex]\\,f\\left(x\\right)=\\frac{x+1}{2-x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135641743\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137565519\">When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id1165137736620\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 2-x& =& 0\\hfill \\\\ \\hfill -x& =& -2\\hfill \\\\ \\hfill x& =& 2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135192763\">Now, we will exclude 2 from the domain. The answers are all real numbers where[latex]\\,x<2\\,[\/latex]or[latex]\\,x>2\\,[\/latex]as shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Image_01_02_028\">(Figure)<\/a>. We can use a symbol known as the union,[latex]\\,\\cup ,[\/latex]to combine the two sets. In interval notation, we write the solution:[latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right).[\/latex]<\/p>\n<div id=\"Image_01_02_028\" class=\"small\">\n<figure style=\"width: 487px\" 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\/19134744\/CNX_Precalc_Figure_01_02_028n.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4. <\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133349280\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137437630\">\n<div id=\"fs-id1165137771815\">\n<p id=\"fs-id1165137442339\">Find the domain of the function:[latex]\\,f\\left(x\\right)=\\frac{1+4x}{2x-1}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137436024\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135186314\">[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135527005\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137733733\"><strong>Given a function written in equation form including an even root, find the domain.<\/strong><\/p>\n<ol id=\"fs-id1165137820030\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for[latex]\\,x.[\/latex]<\/li>\n<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135160109\">\n<div id=\"fs-id1165137735699\">\n<h3>Finding the Domain of a Function with an Even Root<\/h3>\n<p id=\"fs-id1165137466144\">Find the <span class=\"no-emphasis\">domain<\/span> of the function[latex]\\,f\\left(x\\right)=\\sqrt{7-x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137451129\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137453224\">When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.<\/p>\n<p id=\"fs-id1165137749755\">Set the radicand greater than or equal to zero and solve for[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id1165137727831\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 7-x& \\ge & 0\\hfill \\\\ \\hfill -x& \\ge & -7\\hfill \\\\ \\hfill x& \\le & 7\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137422794\">Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to[latex]\\,7,\\,[\/latex]or[latex]\\,\\left(-\\infty ,7\\right].[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737842\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165137933139\">\n<div id=\"fs-id1165137933140\">\n<p id=\"fs-id1165137452448\">Find the domain of the function[latex]\\,f\\left(x\\right)=\\sqrt{5+2x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137832331\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137832332\">[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134328219\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137659456\"><strong>Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\n<p id=\"fs-id1165137937737\"><em>Yes. For example, the function[latex]\\,f\\left(x\\right)=-\\frac{1}{\\sqrt{x}}\\,[\/latex]has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137677916\" class=\"bc-section section\">\n<h3>Using Notations to Specify Domain and Range<\/h3>\n<p id=\"fs-id1165137410091\">In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example,[latex]\\,\\left\\{x|10\\le x<30\\right\\}\\,[\/latex]describes the behavior of[latex]\\,x\\,[\/latex]in set-builder notation. The braces[latex]\\,\\left\\{\\right\\}\\,[\/latex]are read as \u201cthe set of,\u201d and the vertical bar | is read as \u201csuch that,\u201d so we would read[latex]\\,\\left\\{x|10\\le x<30\\right\\}\\,[\/latex]as \u201cthe set of <em>x<\/em>-values such that 10 is less than or equal to[latex]\\,x,\\,[\/latex]and[latex]\\,x\\,[\/latex]is less than 30.\u201d<\/p>\n<p id=\"fs-id1165135207589\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_003\">(Figure)<\/a> compares inequality notation, set-builder notation, and interval notation.<\/p>\n<div id=\"Figure_01_02_003\" 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\/19134751\/CNX_Precalc_Figure_01_02_003.jpg\" alt=\"Summary of notations for inequalities, set-builder, and intervals.\" width=\"975\" height=\"692\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \u201cor.\u201d As we saw in earlier examples, we use the union symbol,[latex]\\,\\cup ,[\/latex]to combine two unconnected intervals. For example, the union of the sets[latex]\\left\\{2,3,5\\right\\}\\,[\/latex]<br \/>\nand[latex]\\,\\left\\{4,6\\right\\}\\,[\/latex]<br \/>\nis the set[latex]\\,\\left\\{2,3,4,5,6\\right\\}.\\,[\/latex]It is the set of all elements that belong to one <em>or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is<\/p>\n<div id=\"fs-id1165135311695\" class=\"unnumbered aligncenter\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/div>\n<div id=\"fs-id1165137641795\" class=\"textbox key-takeaways\">\n<h3>Set-Builder Notation and Interval Notation<\/h3>\n<p id=\"fs-id1165137663670\"><strong>Set-builder notation <\/strong>is a method of specifying a set of elements that satisfy a certain condition. It takes the form[latex]\\left\\{x|\\,\\text{statement about }x\\right\\}\\,[\/latex]which is read as, \u201cthe set of all[latex]\\,x\\,[\/latex]such that the statement about[latex]\\,x\\,[\/latex]is true.\u201d For example,<\/p>\n<div id=\"fs-id1165137543047\" class=\"unnumbered aligncenter\">[latex]\\left\\{x|4<x\\le 12\\right\\}[\/latex]<\/div>\n<p id=\"fs-id1165135190272\"><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,<\/p>\n<div id=\"fs-id1165137443063\" class=\"unnumbered aligncenter\">[latex]\\left(4,12\\right][\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137805770\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137423878\"><strong>Given a line graph, describe the set of values using interval notation.<\/strong><\/p>\n<ol id=\"fs-id1165134032280\" type=\"1\">\n<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\n<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\n<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\n<li>Use the union symbol[latex]\\,\\cup \\,[\/latex]to combine all intervals into one set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165134342702\">\n<div id=\"fs-id1165137803670\">\n<h3>Describing Sets on the Real-Number Line<\/h3>\n<p id=\"fs-id1165137592069\">Describe the intervals of values shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_004\">(Figure)<\/a> using inequality notation, set-builder notation, and interval notation.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/13c74d3bc393a72d003d8fa46c769c591786bc87\/CNX_Precalc_Figure_01_02_004.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135412904\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135412905\">To describe the values,[latex]\\,x,\\,[\/latex]included in the intervals shown, we would say, \u201c[latex]x\\,[\/latex]is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\u201d<\/p>\n<table id=\"fs-id1165137447518\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Inequality<\/strong><\/td>\n<td>[latex]1\\le x\\le 3\\,\\text{or}\\,x>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Set-builder notation<\/strong><\/td>\n<td>[latex]\\left\\{x|1\\le x\\le 3\\,\\text{or}\\,x>5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Interval notation<\/strong><\/td>\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135500794\">Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137779165\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135175087\">\n<p id=\"fs-id1165135341412\">Given <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_005\">(Figure)<\/a>, specify the graphed set in<\/p>\n<ol id=\"fs-id1165137595582\" type=\"a\">\n<li>words<\/li>\n<li>set-builder notation<\/li>\n<li>interval notation<\/li>\n<\/ol>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/eba2f61d5fb32eca2dd57b7de1a1e57511a15f6a\/CNX_Precalc_Figure_01_02_005.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135209390\" class=\"solution textbox shaded\">\n<ol id=\"fs-id1165135528963\" type=\"a\">\n<li>values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3;<\/li>\n<li>[latex]\\left\\{x|x\\le -2\\,\\text{or}\\,-1\\le x<3\\right\\}[\/latex]\n;<\/li>\n<li>[latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653855\" class=\"bc-section section\">\n<h3>Finding Domain and Range from Graphs<\/h3>\n<p id=\"fs-id1165135161404\">Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em>x<\/em>-axis. The range is the set of possible output values, which are shown on the <em>y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_006\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_02_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\/19134810\/CNX_Precalc_Figure_01_02_006.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137597994\">We can observe that the graph extends horizontally from[latex]\\,-5\\,[\/latex]to the right without bound, so the domain is[latex]\\,\\left[-5,\\infty \\right).\\,\\,[\/latex]The vertical extent of the graph is all range values[latex]\\,5\\,[\/latex]and below, so the range is[latex]\\,\\left(\\mathrm{-\\infty },5\\right].\\,[\/latex]Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137561401\">\n<div id=\"fs-id1165137599824\">\n<h3>Finding Domain and Range from a Graph<\/h3>\n<p id=\"fs-id1165135187604\">Find the domain and range of the function[latex]\\,f\\,[\/latex]<br \/>\nwhose graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_02_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\/19134812\/CNX_Precalc_Figure_01_02_007.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137575085\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137768165\">We can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of[latex]\\,f\\,[\/latex]<br \/>\nis[latex]\\,\\left(-3,1\\right].[\/latex]<\/p>\n<p id=\"fs-id1165131968670\">The vertical extent of the graph is 0 to \u20134, so the range is[latex]\\,\\left[-4,0\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_008\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_02_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\/19134818\/CNX_Precalc_Figure_01_02_008.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165134182686\">\n<div id=\"fs-id1165137461643\">\n<h3>Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p id=\"fs-id1165137443324\">Find the domain and range of the function[latex]\\,f\\,[\/latex]whose graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_009\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_02_009\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 489px\" 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\/19134821\/CNX_Precalc_Figure_01_02_009.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11. <\/strong>(credit: modification of work by the U.S. Energy Information Administration)<\/figcaption><\/figure>\n<p><a class=\"footnote\" title=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" id=\"return-footnote-55-2\" href=\"#footnote-55-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137444311\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137476085\">The input quantity along the horizontal axis is \u201cyears,\u201d which we represent with the variable[latex]\\,t\\,[\/latex]for time. The output quantity is \u201cthousands of barrels of oil per day,\u201d which we represent with the variable[latex]\\,b\\,[\/latex]for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as[latex]\\,1973\\le t\\le 2008\\,[\/latex]and the range as approximately[latex]\\,180\\le b\\le 2010.[\/latex]<\/p>\n<p id=\"fs-id1165137747998\">In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545972\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137644581\">\n<p id=\"fs-id1165137644582\">Given <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_010\">(Figure)<\/a>, identify the domain and range using interval notation.<\/p>\n<div id=\"Figure_01_02_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\/19134828\/CNX_Precalc_Figure_01_02_010.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705252\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134079741\">domain =[1950,2002] range = [47,000,000,89,000,000]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434590\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137812796\"><strong>Can a function\u2019s domain and range be the same?<\/strong><\/p>\n<p id=\"fs-id1165137433394\"><em>Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134384565\" class=\"bc-section section\">\n<h3>Finding Domains and Ranges of the Toolkit Functions<\/h3>\n<p id=\"fs-id1165137419914\">We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n<div id=\"Figure_01_02_011\" 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\/19134830\/CNX_Precalc_Figure_01_02_011.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13. <\/strong>For the <strong>constant function<\/strong>[latex]\\,f\\left(x\\right)=c,\\,[\/latex]the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant[latex]\\,c,\\,[\/latex]so the range is the set[latex]\\,\\left\\{c\\right\\}\\,[\/latex]that contains this single element. In interval notation, this is written as[latex]\\,\\left[c,c\\right],\\,[\/latex]the interval that both begins and ends with[latex]\\,c.[\/latex]<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_012\" 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\/19134832\/CNX_Precalc_Figure_01_02_012.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 14.<\/strong> For the <strong>identity function\u2009<\/strong>f(x)=x, there is no restriction on\u2009x. Both the domain and range are the set of all real numbers.<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_013\" 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\/19134844\/CNX_Precalc_Figure_01_02_013.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 15.<\/strong> For the <strong>absolute value function<\/strong>[latex]\\,f\\left(x\\right)=|x|,\\,[\/latex]there is no restriction on[latex]\\,x.\\,[\/latex]However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_014\" 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\/19134904\/CNX_Precalc_Figure_01_02_014.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 16. <\/strong>For the <strong>quadratic function<\/strong>[latex]\\,f\\left(x\\right)={x}^{2},\\,[\/latex]the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_015\" 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\/19134918\/CNX_Precalc_Figure_01_02_015.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 17. <\/strong>For the <strong>cubic function<\/strong>[latex]\\,f\\left(x\\right)={x}^{3},\\,[\/latex]the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_016\" 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\/19134921\/CNX_Precalc_Figure_01_02_016.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 18. <\/strong>For the <strong>reciprocal function<\/strong>[latex]\\,f\\left(x\\right)=\\frac{1}{x},\\,[\/latex]we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write[latex]\\left\\{x|\\text{ }x\\ne 0\\right\\},[\/latex]the set of all real numbers that are not zero.<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_017\" 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\/19134940\/CNX_Precalc_Figure_01_02_017.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 19. <\/strong>For the <strong>reciprocal squared function<\/strong>[latex]\\,f\\left(x\\right)=\\frac{1}{{x}^{2}},[\/latex]we cannot divide by [latex]0,[\/latex] so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_018\" 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\/19134942\/CNX_Precalc_Figure_01_02_018.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 20. <\/strong>For the <strong>square root function<\/strong>[latex]\\,f\\left(x\\right)=\\sqrt[]{x},\\,[\/latex]we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number[latex]\\,x\\,[\/latex]is defined to be positive, even though the square of the negative number[latex]\\,-\\sqrt{x}\\,[\/latex]also gives us[latex]\\,x.[\/latex]<\/figcaption><\/figure>\n<\/div>\n<div id=\"Figure_01_02_019\" 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\/19134947\/CNX_Precalc_Figure_01_02_019.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 21. <\/strong>For the <strong>cube root function<\/strong>[latex]\\,f\\left(x\\right)=\\sqrt[3]{x},\\,[\/latex]the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).<\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165137462732\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137611181\"><strong>Given the formula for a function, determine the domain and range.<\/strong><\/p>\n<ol id=\"fs-id1165137405229\" type=\"1\">\n<li>Exclude from the domain any input values that result in division by zero.<\/li>\n<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\n<li>Use the valid input values to determine the range of the output values.<\/li>\n<li>Look at the function graph and table values to confirm the actual function behavior.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137558723\">\n<div id=\"fs-id1165137464274\">\n<h3>Finding the Domain and Range Using Toolkit Functions<\/h3>\n<p id=\"fs-id1165135613224\">Find the domain and range of[latex]\\,f\\left(x\\right)=2{x}^{3}-x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135458670\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137527861\">There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.<\/p>\n<p id=\"fs-id1165135208585\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]and the range is also[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137448155\">\n<div id=\"fs-id1165137661316\">\n<h3>Finding the Domain and Range<\/h3>\n<p id=\"fs-id1165137419507\">Find the domain and range of[latex]\\,f\\left(x\\right)=\\frac{2}{x+1}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137871182\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137855321\">We cannot evaluate the function at[latex]\\,-1\\,[\/latex]because division by zero is undefined. The domain is[latex]\\,\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right).\\,[\/latex]Because the function is never zero, we exclude 0 from the range. The range is[latex]\\,\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137574740\">\n<div id=\"fs-id1165135641583\">\n<h3>Finding the Domain and Range<\/h3>\n<p id=\"fs-id1165137661054\">Find the domain and range of[latex]\\,f\\left(x\\right)=2\\sqrt{x+4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137584342\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137596350\">We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\n<div id=\"eip-id1165137567088\" class=\"unnumbered\">[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/div>\n<p id=\"fs-id1165137465335\">The domain of[latex]\\,f\\left(x\\right)\\,[\/latex]is[latex]\\,\\left[-4,\\infty \\right).[\/latex]<\/p>\n<p id=\"fs-id1165137544393\">We then find the range. We know that[latex]\\,f\\left(-4\\right)=0,\\,[\/latex]and the function value increases as[latex]\\,x\\,[\/latex]increases without any upper limit. We conclude that the range of[latex]\\,f\\,[\/latex]is[latex]\\,\\left[0,\\infty \\right).[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137572635\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137437183\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_020\">(Figure)<\/a> represents the function[latex]\\,f.[\/latex]<\/p>\n<div id=\"Figure_01_02_020\" 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\/19134949\/CNX_Precalc_Figure_01_02_020.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 22.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137430800\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137475544\">\n<p id=\"fs-id1165137475545\">Find the domain and range of[latex]\\,f\\left(x\\right)=-\\sqrt{2-x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137833252\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137725047\">domain:[latex]\\,\\left(-\\infty ,2\\right];\\,[\/latex]range:[latex]\\,\\left(-\\infty ,0\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440477\" class=\"bc-section section\">\n<h3>Graphing Piecewise-Defined Functions<\/h3>\n<p id=\"fs-id1165137409262\">Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function[latex]\\,f\\left(x\\right)=|x|.\\,[\/latex]With a domain of all real numbers and a range of values greater than or equal to 0, <span class=\"no-emphasis\">absolute value<\/span> can be defined as the <span class=\"no-emphasis\">magnitude<\/span>, or <span class=\"no-emphasis\">modulus<\/span>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\n<p id=\"fs-id1165137558775\">If we input 0, or a positive value, the output is the same as the input.<\/p>\n<div id=\"fs-id1165135194329\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=x\\,\\text{if}\\,x\\ge 0[\/latex]<\/div>\n<p id=\"fs-id1165137529947\">If we input a negative value, the output is the opposite of the input.<\/p>\n<div id=\"fs-id1165133112779\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=-x\\,\\text{if}\\,x<0[\/latex]<\/div>\n<p id=\"fs-id1165137863778\">Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\n<p id=\"fs-id1165134042316\">We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income[latex]\\,S\\,[\/latex]would be[latex]\\,0.1S\\,[\/latex]if[latex]\\,S\\le \\text{\\$}10\\text{,}000\\,[\/latex]and[latex]\\,\\text{\\$}1000+0.2\\left(S-\\text{\\$}10\\text{,}000\\right)\\,[\/latex]if[latex]\\,S>\\text{\\$}10\\text{,}000.[\/latex]<\/p>\n<div id=\"fs-id1165137531241\" class=\"textbox key-takeaways\">\n<h3>Piecewise Function<\/h3>\n<p id=\"fs-id1165135504970\">A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:<\/p>\n<div id=\"fs-id1165137482244\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=\\Bigg\\{\\begin{array}{l}\\text{formula 1 if }x\\text{ is in domain 1}\\\\ \\text{formula 2 if }x\\text{ is in domain 2}\\\\ \\text{formula 3 if }x\\text{ is in domain 3}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137543841\">In piecewise notation, the absolute value function is<\/p>\n<div id=\"fs-id1165135190749\" class=\"unnumbered aligncenter\">[latex]|x|=\\bigg\\{\\begin{array}{l}x\\text{ if }x\\ge 0\\\\ -x\\text{ if }x<0\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137768426\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137823161\"><strong>Given a piecewise function, write the formula and identify the domain for each interval.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165135443772\" type=\"1\">\n<li>Identify the intervals for which different rules apply.<\/li>\n<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\n<li>Use braces and if-statements to write the function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137452506\">\n<div id=\"fs-id1165135321994\">\n<h3>Writing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137834905\">A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <span class=\"no-emphasis\">function<\/span> relating the number of people,[latex]\\,n,\\,[\/latex]to the cost,[latex]\\,C.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137807421\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135331729\">Two different formulas will be needed. For <em>n<\/em>-values under 10,[latex]\\,C=5n.\\,[\/latex]For values of[latex]\\,n\\,[\/latex]that are 10 or greater,[latex]\\,C=50.[\/latex]<\/p>\n<div id=\"fs-id1165135208951\" class=\"unnumbered aligncenter\">[latex]C\\left(n\\right)=\\left\\{\\begin{array}{ccc}5n& \\text{if}& 0<n<10\\\\ 50& \\text{if}& n\\ge 10\\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135436578\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135196985\">The function is represented in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_021\">(Figure)<\/a>. The graph is a diagonal line from[latex]\\,n=0\\,[\/latex]to[latex]\\,n=10\\,[\/latex]and a constant after that. In this example, the two formulas agree at the meeting point where[latex]\\,n=10,\\,[\/latex]but not all piecewise functions have this property.<\/p>\n<div id=\"Figure_01_02_021\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 360px\" 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\/19134959\/CNX_Precalc_Figure_01_02_021.jpg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 23.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135436662\">\n<div id=\"fs-id1165135436664\">\n<h3>Working with a Piecewise Function<\/h3>\n<p id=\"fs-id1165137938645\">A cell phone company uses the function below to determine the cost,[latex]\\,C,\\,[\/latex]in dollars for[latex]\\,g\\,[\/latex]gigabytes of data transfer.<\/p>\n<div id=\"fs-id1165137660470\" class=\"unnumbered aligncenter\">[latex]C\\left(g\\right)=\\left\\{\\begin{array}{ccc}25& \\text{if}& 0<g<2\\\\ 25+10\\left(g-2\\right)& \\text{if}& g\\ge 2\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135193798\">Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.<\/p>\n<\/div>\n<div id=\"fs-id1165135177567\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134373545\">To find the cost of using 1.5 gigabytes of data,[latex]\\,C\\left(1.5\\right),\\,[\/latex]we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.<\/p>\n<div id=\"fs-id1165134300204\" class=\"unnumbered aligncenter\">[latex]C\\left(1.5\\right)=\\text{\\$}25[\/latex]<\/div>\n<p id=\"fs-id1165135440213\">To find the cost of using 4 gigabytes of data,[latex]\\,C\\left(4\\right),\\,[\/latex]we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\n<div id=\"fs-id1165135383665\" class=\"unnumbered aligncenter\">[latex]C\\left(4\\right)=25+10\\left(4-2\\right)=\\text{\\$}45[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634432\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137601265\">The function is represented in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_022\">(Figure)<\/a>. We can see where the function changes from a constant to a shifted and stretched identity at[latex]\\,g=2.\\,[\/latex]We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.<\/p>\n<div id=\"Figure_01_02_022\" 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\/19135002\/CNX_Precalc_Figure_01_02_022.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 24.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600493\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135532516\"><strong>Given a piecewise function, sketch a graph.<\/strong><\/p>\n<ol id=\"fs-id1165137588539\" type=\"1\">\n<li>Indicate on the <em>x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\n<li>For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137781618\">\n<div id=\"fs-id1165135412870\">\n<h3>Graphing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137838785\">Sketch a graph of the function.<\/p>\n<div id=\"fs-id1165137475346\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}{x}^{2}& \\text{if}& x\\le 1\\\\ 3& \\text{if}& 1<x\\le 2\\\\ x& \\text{if}& x>2\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135487148\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135487150\">Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.<\/p>\n<p id=\"fs-id1165137642848\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_023\">(Figure)<\/a> shows the three components of the piecewise function graphed on separate coordinate systems.<\/p>\n<div id=\"Figure_01_02_023\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 974px\" 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\/19135010\/CNX_Precalc_Figure_01_02_023abc.jpg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 25. <\/strong>(a)[latex]\\,f\\left(x\\right)={x}^{2}\\text{ if }x\\le 1;\\,[\/latex](b)[latex]\\,f\\left(x\\right)=3\\text{ if 1&lt; }x\\le 2;\\,[\/latex](c)[latex]\\,f\\left(x\\right)=x\\text{ if }x&gt;2[\/latex]<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137676209\">Now that we have sketched each piece individually, we combine them in the same coordinate plane. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Figure_01_02_026\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_02_026\" 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\/19135018\/CNX_Precalc_Figure_01_02_026.jpg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 26.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135188517\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134389893\">Note that the graph does pass the vertical line test even at[latex]\\,x=1\\,[\/latex]and[latex]\\,x=2\\,[\/latex]because the points [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(2,2\\right)[\/latex] are not part of the graph of the function, though [latex]\\left(1,1\\right)[\/latex]<br \/>\nand [latex]\\left(2,\\,3\\right)[\/latex] are.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762558\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137692562\">\n<p id=\"fs-id1165137692563\">Graph the following piecewise function.<\/p>\n<div id=\"fs-id1165137433350\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}{x}^{3}& \\text{if}& x<-1\\\\ -2& \\text{if}& -1<x<4\\\\ \\sqrt{x}& \\text{if}& x>4\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137784656\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165134302462\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135020\/CNX_Precalc_Figure_01_02_027.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810682\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137527804\"><strong>Can more than one formula from a piecewise function be applied to a value in the domain?<\/strong><\/p>\n<p id=\"fs-id1165137464467\"><em>No. Each value corresponds to one equation in a piecewise formula.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165135190393\" class=\"precalculus media\">\n<p id=\"fs-id1165137627040\">Access these online resources for additional instruction and practice with domain and range.<\/p>\n<ul id=\"fs-id1165135189954\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/domainsqroot\">Domain and Range of Square Root Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/determinedomain\">Determining Domain and Range<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/drgraph\">Find Domain and Range Given the Graph<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/drtable\">Find Domain and Range Given a Table<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/drcoordinate\">Find Domain and Range Given Points on a Coordinate Plane<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134077347\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137591772\">\n<li>The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.<\/li>\n<li>The domain of a function can be determined by listing the input values of a set of ordered pairs. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_01\">(Figure)<\/a>.<\/li>\n<li>The domain of a function can also be determined by identifying the input values of a function written as an equation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_02\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_03\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_04\">(Figure)<\/a>.<\/li>\n<li>Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_05\">(Figure)<\/a>.<\/li>\n<li>For many functions, the domain and range can be determined from a graph. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_07\">(Figure)<\/a>.<\/li>\n<li>An understanding of toolkit functions can be used to find the domain and range of related functions. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_08\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_09\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_10\">(Figure)<\/a>.<\/li>\n<li>A piecewise function is described by more than one formula. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_11\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_12\">(Figure)<\/a>.<\/li>\n<li>A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2265&amp;action=edit#Example_01_02_13\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135176628\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135172218\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137665109\">\n<div id=\"fs-id1165135245908\">\n<p id=\"fs-id1165135245910\">Why does the domain differ for different functions?<\/p>\n<\/div>\n<div id=\"fs-id1165134199600\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135613709\">The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440209\">\n<div id=\"fs-id1165135533141\">\n<p id=\"fs-id1165135533143\">How do we determine the domain of a function defined by an equation?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635386\">\n<div id=\"fs-id1165135390940\">\n<p id=\"fs-id1165135390942\">Explain why the domain of[latex]\\,f\\left(x\\right)=\\sqrt[3]{x}\\,[\/latex]is different from the domain of[latex]\\,f\\left(x\\right)=\\sqrt[]{x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137727146\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137727148\">There is no restriction on[latex]\\,x\\,[\/latex]for[latex]\\,f\\left(x\\right)=\\sqrt[3]{x}\\,[\/latex]because you can take the cube root of any real number. So the domain is all real numbers,[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]When dealing with the set of real numbers, you cannot take the square root of negative numbers. So[latex]\\,x[\/latex]-values are restricted for[latex]\\,f\\left(x\\right)=\\sqrt[]{x}\\,[\/latex]to nonnegative numbers and the domain is[latex]\\,\\left[0,\\infty \\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042454\">\n<div id=\"fs-id1165134042457\">\n<p id=\"fs-id1165137438149\">When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134211324\">\n<div id=\"fs-id1165137446310\">\n<p id=\"fs-id1165137446313\">How do you graph a piecewise function?<\/p>\n<\/div>\n<div id=\"fs-id1165137574335\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135415726\">Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the[latex]\\,x[\/latex]-axis and[latex]\\,y[\/latex]-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate[latex]\\,-\\infty \\,[\/latex]or[latex]\\,\\text{ }\\infty .\\,[\/latex]Combine the graphs to find the graph of the piecewise function.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771069\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137408926\">For the following exercises, find the domain of each function using interval notation.<\/p>\n<div id=\"fs-id1165137833819\">\n<div id=\"fs-id1165137833821\">\n<p id=\"fs-id1165135500745\">[latex]f\\left(x\\right)=-2x\\left(x-1\\right)\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137854912\">\n<div id=\"fs-id1165134312130\">\n<p id=\"fs-id1165134312132\">[latex]f\\left(x\\right)=5-2{x}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137731586\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137731589\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135512534\">\n<div id=\"fs-id1165135512537\">\n<p id=\"fs-id1165137804475\">[latex]f\\left(x\\right)=3\\sqrt{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473385\">\n<div id=\"fs-id1165137473388\">\n<p id=\"fs-id1165134374059\">[latex]f\\left(x\\right)=3-\\sqrt{6-2x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137451053\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137451055\">[latex]\\left(-\\infty ,3\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192268\">\n<div id=\"fs-id1165135192270\">\n<p id=\"fs-id1165137725224\">[latex]f\\left(x\\right)=\\sqrt{4-3x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137629066\">\n<div id=\"fs-id1165137483196\">\n<p id=\"fs-id1165137483198\">[latex]\\begin{array}{l}\\\\ f\\left(x\\right)=\\sqrt[]{{x}^{2}+4}\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134192936\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137463777\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137807107\">\n<div id=\"fs-id1165137551129\">\n<p id=\"fs-id1165137551131\">[latex]f\\left(x\\right)=\\sqrt[3]{1-2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134259277\">\n<div id=\"fs-id1165134259279\">\n<p id=\"fs-id1165135186001\">[latex]f\\left(x\\right)=\\sqrt[3]{x-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134058389\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137836596\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503751\">\n<div id=\"fs-id1165134156030\">\n<p id=\"fs-id1165134156032\">[latex]f\\left(x\\right)=\\frac{9}{x-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133276237\">\n<div id=\"fs-id1165133276240\">\n<p id=\"fs-id1165137784864\">[latex]f\\left(x\\right)=\\frac{3x+1}{4x+2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137454548\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134170171\">[latex]\\left(-\\infty ,-\\frac{1}{2}\\right)\\cup \\left(-\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810520\">\n<div id=\"fs-id1165137810522\">\n<p id=\"fs-id1165137532795\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x+4}}{x-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135548992\">\n<div id=\"fs-id1165135634123\">\n<p id=\"fs-id1165135634125\">[latex]f\\left(x\\right)=\\frac{x-3}{{x}^{2}+9x-22}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137528909\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137528911\">[latex]\\left(-\\infty ,-11\\right)\\cup \\left(-11,2\\right)\\cup \\left(2,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593402\">\n<div id=\"fs-id1165135593404\">\n<p id=\"fs-id1165137771850\">[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-x-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191342\">\n<div id=\"fs-id1165134284474\">\n<p id=\"fs-id1165134284476\">[latex]f\\left(x\\right)=\\frac{2{x}^{3}-250}{{x}^{2}-2x-15}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135256053\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135256055\">[latex]\\left(-\\infty ,-3\\right)\\cup \\left(-3,5\\right)\\cup \\left(5,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137921795\">\n<div id=\"fs-id1165137921797\">\n<p id=\"fs-id1165137532172\">[latex]\\frac{5}{\\sqrt{x-3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476914\">\n<div id=\"fs-id1165137476916\">\n<p id=\"fs-id1165137726504\">[latex]\\frac{2x+1}{\\sqrt{5-x}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137647829\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137564959\">[latex]\\left(-\\infty ,5\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135185292\">\n<div id=\"fs-id1165137640755\">\n<p id=\"fs-id1165137640757\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x-4}}{\\sqrt{x-6}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135252252\">\n<div id=\"fs-id1165137611840\">\n<p id=\"fs-id1165137611842\">[latex]f\\left(x\\right)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137611238\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137538970\">[latex]\\left[6,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601712\">\n<div id=\"fs-id1165137601714\">\n<p id=\"fs-id1165137657487\">[latex]f\\left(x\\right)=\\frac{x}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628472\">\n<div id=\"fs-id1165137651574\">\n<p id=\"fs-id1165137651576\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-9x}{{x}^{2}-81}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135188135\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137809882\">[latex]\\left(-\\infty ,-9\\right)\\cup \\left(-9,9\\right)\\cup \\left(9,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137469452\">\n<div id=\"fs-id1165137469454\">\n<p id=\"fs-id1165137635293\">Find the domain of the function[latex]\\,f\\left(x\\right)=\\sqrt{2{x}^{3}-50x}\\,[\/latex]by:<\/p>\n<ol id=\"fs-id1165137938832\" type=\"a\">\n<li>using algebra.<\/li>\n<li>graphing the function in the radicand and determining intervals on the <em>x<\/em>-axis for which the radicand is nonnegative.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137580833\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135186809\">For the following exercises, write the domain and range of each function using interval notation.<\/p>\n<div id=\"fs-id1165135168172\">\n<div id=\"fs-id1165137647479\"><span id=\"fs-id1165137891294\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135032\/CNX_Precalc_Figure_01_02_202.jpg\" alt=\"Graph of a function from (2, 8].\" \/><\/span><\/div>\n<div id=\"fs-id1165137820038\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137424631\">domain:[latex]\\,\\left(2,8\\right],\\,[\/latex]range[latex]\\,\\left[6,8\\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160181\">\n<div id=\"fs-id1165135160183\"><span id=\"fs-id1165137837830\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135034\/CNX_Precalc_Figure_01_02_203.jpg\" alt=\"Graph of a function from [4, 8).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137723404\">\n<div id=\"fs-id1165137809982\"><span id=\"fs-id1165137733767\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135035\/CNX_Precalc_Figure_01_02_204.jpg\" alt=\"Graph of a function from [-4, 4].\" \/><\/span><\/div>\n<div id=\"fs-id1165137847285\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137541038\">domain:[latex]\\,\\left[-4,\\text{ 4],}\\,[\/latex]range:[latex]\\,\\left[0,\\text{ 2]}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137590678\">\n<div id=\"fs-id1165134168421\"><span id=\"fs-id1165137837060\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135045\/CNX_Precalc_Figure_01_02_205.jpg\" alt=\"Graph of a function from [2, 6].\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137737326\">\n<div id=\"fs-id1165137737328\"><span id=\"fs-id1165134129572\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135058\/CNX_Precalc_Figure_01_02_206.jpg\" alt=\"Graph of a function from [-5, 3).\" \/><\/span><\/div>\n<div id=\"fs-id1165137657479\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137657482\">domain:[latex]\\,\\left[-5,\\text{ }3\\right),\\,[\/latex]range:[latex]\\,\\left[0,2\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404973\">\n<div id=\"fs-id1165137404975\"><span id=\"fs-id1165134305418\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135100\/CNX_Precalc_Figure_01_02_207.jpg\" alt=\"Graph of a function from [-3, 2).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137544188\">\n<div id=\"fs-id1165137437269\"><span id=\"fs-id1165137447903\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135103\/CNX_Precalc_Figure_01_02_208.jpg\" alt=\"Graph of a function from (-infinity, 2].\" \/><\/span><\/div>\n<div id=\"fs-id1165137445711\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137445713\">domain:[latex]\\,\\left(-\\infty ,1\\right],\\,[\/latex]range:[latex]\\,\\left[0,\\infty \\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176309\">\n<div id=\"fs-id1165134323791\"><span id=\"fs-id1165135192955\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135105\/CNX_Precalc_Figure_01_02_209.jpg\" alt=\"Graph of a function from [-4, infinity).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137642580\">\n<div id=\"fs-id1165137642582\"><span id=\"fs-id1165134482733\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135116\/CNX_Precalc_Figure_01_02_210.jpg\" alt=\"Graph of a function from [-6, -1\/6]U[1\/6, 6]\/.\" \/><\/span><\/div>\n<div id=\"fs-id1165134043582\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135335983\">domain:[latex]\\,\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right];\\,[\/latex]range:[latex]\\,\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right]\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137442385\">\n<div id=\"fs-id1165137812572\"><span id=\"fs-id1165137645308\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135118\/CNX_Precalc_Figure_01_02_211.jpg\" alt=\"Graph of a function from (-2.5, infinity).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137851981\">\n<div id=\"fs-id1165137851983\"><span id=\"fs-id1165137602824\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135121\/CNX_Precalc_Figure_01_02_212.jpg\" alt=\"Graph of a function from [-3, infinity).\" \/><\/span><\/div>\n<div id=\"fs-id1165137575572\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137601170\">domain:[latex]\\,\\left[-3,\\text{ }\\infty \\right);\\,[\/latex]range:[latex]\\,\\left[0,\\infty \\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137785119\">For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\n<div id=\"fs-id1165137462167\">\n<div id=\"fs-id1165137408525\">\n<p id=\"fs-id1165137408527\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}x+1\\hfill & \\text{if}\\hfill & x<-2\\hfill \\\\ -2x-3\\hfill & \\text{if}\\hfill & x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137562309\">\n<div id=\"fs-id1165134328320\">\n<p id=\"fs-id1165134328322\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}2x-1\\hfill & \\text{if}\\hfill & x<1\\hfill \\\\ 1+x\\hfill & \\text{if}\\hfill & x\\ge 1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135481131\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135481133\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<p><span id=\"fs-id1165137662700\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135123\/CNX_Precalc_Figure_01_02_214.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628033\">\n<div id=\"fs-id1165137658060\">\n<p id=\"fs-id1165137658062\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}x+1\\,\\,\\text{if}\\,\\,x<0\\\\ x-1\\,\\,\\text{if}\\,\\,\\,x>0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135641679\">\n<div id=\"fs-id1165135641681\">\n<p id=\"fs-id1165133402089\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}3& \\text{if}& x<0\\\\ \\sqrt{x}& \\text{if}& x\\ge 0\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137500956\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135532432\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<p><span id=\"fs-id1165137474386\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135125\/CNX_Precalc_Figure_01_02_216.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192719\">\n<div id=\"fs-id1165135192721\">\n<p id=\"fs-id1165137400953\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}{x}^{2}\\text{ if }x<0\\\\ 1-x\\text{ if }x>0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137594981\">\n<div id=\"fs-id1165135210029\">\n<p id=\"fs-id1165135210031\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{r}\\hfill \\begin{array}{r}\\hfill {x}^{2}\\\\ \\hfill x+2\\end{array}\\end{array}\\,\\,\\begin{array}{l}\\text{if}\\,\\,\\,\\,\\,x<0\\hfill \\\\ \\text{if}\\,\\,\\,\\,\\,x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137667233\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135382142\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<p><span id=\"fs-id1165135188662\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135134\/CNX_Precalc_Figure_01_02_218.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137571389\">\n<div id=\"fs-id1165137433000\">\n<p id=\"fs-id1165137433002\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}x+1& \\text{if}& x<1\\\\ {x}^{3}& \\text{if}& x\\ge 1\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137407891\">\n<div id=\"fs-id1165137554125\">\n<p id=\"fs-id1165137554127\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}|x|\\\\ 1\\end{array}\\begin{array}{l}\\,\\,\\,\\text{if}\\,\\,\\,x<2\\hfill \\\\ \\,\\,\\,\\text{if}\\,\\,\\,x\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137401041\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134252896\">domain:[latex]\\,\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<p><span id=\"fs-id1165135432997\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135140\/CNX_Precalc_Figure_01_02_220.jpg\" alt=\"Graph of f(x).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134118450\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165135188383\">For the following exercises, given each function [latex]f,[\/latex]evaluate [latex]f\\left(-3\\right),\\,f\\left(-2\\right),\\,f\\left(-1\\right),[\/latex] and [latex]f\\left(0\\right).[\/latex]<\/p>\n<div id=\"fs-id1165137471865\">\n<div id=\"fs-id1165137471867\">\n<p id=\"fs-id1165134043731\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}x+1\\hfill & \\text{if}\\hfill & x<-2\\hfill \\\\ -2x-3\\hfill & \\text{if}\\hfill & x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122954\">\n<div id=\"fs-id1165134122956\">\n<p id=\"fs-id1165135168423\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{cc}1& \\text{if }x\\le -3\\\\ 0& \\text{if }x>-3\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137804494\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137804496\">[latex]\\begin{array}{cccc}f\\left(-3\\right)=1;& f\\left(-2\\right)=0;& f\\left(-1\\right)=0;& f\\left(0\\right)=0\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556768\">\n<div id=\"fs-id1165137423742\">\n<p id=\"fs-id1165137423744\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{cc}-2{x}^{2}+3& \\text{if }x\\le -1\\\\ 5x-7& \\text{if }x>-1\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137469026\">For the following exercises, given each function[latex]\\,f,\\,[\/latex]evaluate[latex]f\\left(-1\\right),\\,f\\left(0\\right),\\,f\\left(2\\right),\\,[\/latex]and[latex]\\,f\\left(4\\right).[\/latex]<\/p>\n<div>\n<div id=\"fs-id1165134380353\">\n<p id=\"fs-id1165137678245\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{lll}7x+3\\hfill & \\text{if}\\hfill & x<0\\hfill \\\\ 7x+6\\hfill & \\text{if}\\hfill & x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137476514\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137476516\">[latex]\\begin{array}{cccc}f\\left(-1\\right)=-4;& f\\left(0\\right)=6;& f\\left(2\\right)=20;& f\\left(4\\right)=34\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137693713\">\n<div id=\"fs-id1165137679373\">\n<p id=\"fs-id1165137679375\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}{x}^{2}-2& \\text{if}& x<2\\\\ 4+|x-5|& \\text{if}& x\\ge 2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137715004\">\n<div id=\"fs-id1165137715006\">\n<p id=\"fs-id1165137715008\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{ccc}5x& \\text{if}& x<0\\\\ 3& \\text{if}& 0\\le x\\le 3\\\\ {x}^{2}& \\text{if}& x>3\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135699157\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137401550\">[latex]\\begin{array}{cccc}f\\left(-1\\right)=-5;& f\\left(0\\right)=3;& f\\left(2\\right)=3;& f\\left(4\\right)=16\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137837869\">For the following exercises, write the domain for the piecewise function in interval notation.<\/p>\n<div id=\"fs-id1165137837872\">\n<div id=\"fs-id1165135341427\">\n<p id=\"fs-id1165135341429\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}x+1\\,\\,\\,\\,\\,\\text{ if}\\,\\,x<-2\\\\ -2x-3\\,\\,\\text{if}\\,\\,x\\ge -2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704661\">\n<div id=\"fs-id1165137704664\">\n<p id=\"fs-id1165137704666\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}{x}^{2}-2\\,\\,\\,\\,\\,\\text{ if}\\,\\,x<1\\\\ -{x}^{2}+2\\,\\,\\text{if}\\,\\,x>1\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135420410\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135570357\">domain:[latex]\\,\\left(-\\infty ,1\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137772429\">\n<div id=\"fs-id1165137772431\">\n<p id=\"fs-id1165137675983\">[latex]f\\left(x\\right)=\\left\\{\\begin{array}{c}2x-3\\\\ -3{x}^{2}\\end{array}\\,\\,\\begin{array}{c}\\text{if}\\,\\,\\,x<0\\\\ \\text{if}\\,\\,\\,x\\ge 2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135194497\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165137780865\">\n<div id=\"fs-id1165137780867\">\n<p id=\"fs-id1165135641711\">Graph[latex]\\,y=\\frac{1}{{x}^{2}}\\,[\/latex]on the viewing window[latex]\\,\\left[-0.5,-0.1\\right]\\,[\/latex]and[latex]\\,\\left[0.1,0.5\\right].\\,[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n<\/div>\n<div id=\"fs-id1165137501974\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/e9ff86fef34aa7f3335b023686c29d96c31810c9\/CNX_Precalc_Figure_01_02_221.jpg\" alt=\"Graph of the equation from [-0.5, -0.1].\" \/><\/p>\n<p id=\"fs-id1165135191028\">window:<span id=\"MathJax-Element-2804-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51126\" class=\"math\"><span id=\"MathJax-Span-51127\" class=\"mrow\"><span id=\"MathJax-Span-51128\" class=\"semantics\"><span id=\"MathJax-Span-51129\" class=\"mrow\"><span id=\"MathJax-Span-51130\" class=\"mrow\"><span id=\"MathJax-Span-51131\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51132\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51133\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-51134\" class=\"mn\">0.5<\/span><span id=\"MathJax-Span-51135\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51136\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-51137\" class=\"mn\">0.1<\/span><span id=\"MathJax-Span-51138\" class=\"mo\">]<\/span><span id=\"MathJax-Span-51139\" class=\"mo\">;<\/span><span id=\"MathJax-Span-51140\" class=\"mtext\">\u2009<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[\u22120.5,\u22120.1];\u2009\u2009[\u22120.5,\u22120.1];\u2009<\/span><\/span> range:<span id=\"MathJax-Element-2805-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51141\" class=\"math\"><span id=\"MathJax-Span-51142\" class=\"mrow\"><span id=\"MathJax-Span-51143\" class=\"semantics\"><span id=\"MathJax-Span-51144\" class=\"mrow\"><span id=\"MathJax-Span-51145\" class=\"mrow\"><span id=\"MathJax-Span-51146\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51147\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51148\" class=\"mn\">4<\/span><span id=\"MathJax-Span-51149\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51150\" class=\"mtext\">&nbsp;<\/span><span id=\"MathJax-Span-51151\" class=\"mn\">100<\/span><span id=\"MathJax-Span-51152\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[4,&nbsp;100]\u2009[4,&nbsp;100]<\/span><\/span><\/p>\n<p><span id=\"fs-id1165137442910\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/1958a0a7420a22e38d7733e4c3481051536b1e52\/CNX_Precalc_Figure_01_02_222.jpg\" alt=\"Graph of the equation from [0.1, 0.5].\" \/> <\/span><\/p>\n<p id=\"fs-id1165134378637\">window:<span id=\"MathJax-Element-2806-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51153\" class=\"math\"><span id=\"MathJax-Span-51154\" class=\"mrow\"><span id=\"MathJax-Span-51155\" class=\"semantics\"><span id=\"MathJax-Span-51156\" class=\"mrow\"><span id=\"MathJax-Span-51157\" class=\"mrow\"><span id=\"MathJax-Span-51158\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51159\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51160\" class=\"mn\">0.1<\/span><span id=\"MathJax-Span-51161\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51162\" class=\"mtext\">&nbsp;<\/span><span id=\"MathJax-Span-51163\" class=\"mn\">0.5<\/span><span id=\"MathJax-Span-51164\" class=\"mo\">]<\/span><span id=\"MathJax-Span-51165\" class=\"mo\">;<\/span><span id=\"MathJax-Span-51166\" class=\"mtext\">\u2009<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[0.1,&nbsp;0.5];\u2009\u2009[0.1,&nbsp;0.5];\u2009<\/span><\/span> range:<span id=\"MathJax-Element-2807-Frame\" class=\"MathJax\" role=\"presentation\"><span id=\"MathJax-Span-51167\" class=\"math\"><span id=\"MathJax-Span-51168\" class=\"mrow\"><span id=\"MathJax-Span-51169\" class=\"semantics\"><span id=\"MathJax-Span-51170\" class=\"mrow\"><span id=\"MathJax-Span-51171\" class=\"mrow\"><span id=\"MathJax-Span-51172\" class=\"mtext\">\u2009<\/span><span id=\"MathJax-Span-51173\" class=\"mo\">[<\/span><span id=\"MathJax-Span-51174\" class=\"mn\">4<\/span><span id=\"MathJax-Span-51175\" class=\"mo\">,<\/span><span id=\"MathJax-Span-51176\" class=\"mtext\">&nbsp;<\/span><span id=\"MathJax-Span-51177\" class=\"mn\">100<\/span><span id=\"MathJax-Span-51178\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span>&nbsp;<\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u2009[4,&nbsp;100]\u2009[4,&nbsp;100]<\/span><\/span><\/p>\n<p id=\"fs-id1165134378637\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131911953\">\n<div id=\"fs-id1165137842479\">\n<p id=\"fs-id1165137842481\">Graph[latex]\\,y=\\frac{1}{x}\\,[\/latex]on the viewing window[latex]\\,\\left[-0.5,-0.1\\right]\\,[\/latex]and[latex]\\,\\left[0.1,\\text{ }0.5\\right].\\,[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137733672\" class=\"bc-section section\">\n<h4>Extension<\/h4>\n<div id=\"fs-id1165137442197\">\n<div id=\"fs-id1165133221851\">\n<p id=\"fs-id1165133221853\">Suppose the range of a function[latex]\\,f\\,[\/latex]is[latex]\\,\\left[-5,\\text{ }8\\right].\\,[\/latex]What is the range of[latex]\\,|f\\left(x\\right)|?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134555582\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134555584\">[latex]\\left[0,\\text{ }8\\right][\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137679047\">\n<div id=\"fs-id1165137679049\">\n<p id=\"fs-id1165133410011\">Create a function in which the range is all nonnegative real numbers.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209378\">\n<div id=\"fs-id1165135209380\">\n<p id=\"fs-id1165137645593\">Create a function in which the domain is[latex]\\,x>2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133210812\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137779064\">Many answers. One function is[latex]\\,f\\left(x\\right)=\\frac{1}{\\sqrt{x-2}}.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832031\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135511303\">\n<div id=\"fs-id1165135511305\">\n<p id=\"fs-id1165135336103\">The height[latex]\\,h\\,[\/latex]of a projectile is a function of the time[latex]\\,t\\,[\/latex]it is in the air. The height in feet for[latex]\\,t\\,[\/latex]seconds is given by the function[latex]h\\left(t\\right)=-16{t}^{2}+96t.[\/latex]<br \/>\nWhat is the domain of the function? What does the domain mean in the context of the problem?<\/p>\n<\/div>\n<div id=\"fs-id1165137446701\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137758760\">The domain is[latex]\\,\\left[0,\\text{ }6\\right];\\,[\/latex]it takes 6 seconds for the projectile to leave the ground and return to the ground<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137406705\">\n<div id=\"fs-id1165137406708\">\n<p id=\"fs-id1165133045371\">The cost in dollars of making[latex]\\,x\\,[\/latex]items is given by the function[latex]\\,C\\left(x\\right)=10x+500.[\/latex]<\/p>\n<ol id=\"fs-id1165137862357\" type=\"a\">\n<li>The fixed cost is determined when zero items are produced. Find the fixed cost for this item.<\/li>\n<li>What is the cost of making 25 items?<\/li>\n<li>Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function,[latex]\\,C\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl>\n<dt>interval notation<\/dt>\n<dd id=\"fs-id1165135190252\">a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135487256\">\n<dt>piecewise function<\/dt>\n<dd id=\"fs-id1165137452169\">a function in which more than one formula is used to define the output<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137863188\">\n<dt>set-builder notation<\/dt>\n<dd id=\"fs-id1165137863193\">a method of describing a set by a rule that all of its members obey; it takes the form[latex]\\,\\left\\{x|\\,\\text{statement about }x\\right\\}[\/latex]<\/dd>\n<\/dl>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-55-1\">The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014 <a href=\"#return-footnote-55-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-55-2\">http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A. <a href=\"#return-footnote-55-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":291,"menu_order":3,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-55","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/55","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\/55\/revisions"}],"predecessor-version":[{"id":56,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/55\/revisions\/56"}],"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\/55\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=55"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=55"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=55"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=55"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}