{"id":13664,"date":"2018-08-23T18:15:01","date_gmt":"2018-08-23T18:15:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13664"},"modified":"2020-05-21T04:35:50","modified_gmt":"2020-05-21T04:35:50","slug":"domain-and-range","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/domain-and-range\/","title":{"raw":"Section 1.6: Domain and Range","rendered":"Section 1.6: Domain and Range"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the domain of a function defined by an equation.<\/li>\r\n \t<li>Find the domain of a function from its graph.<\/li>\r\n \t<li>Graph piecewise-defined functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\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>. Figure 1\u00a0shows 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 <strong>domain<\/strong> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_0012.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\" \/> <b>Figure 1.<\/b> Based on data compiled by <a href=\"http:\/\/www.the-numbers.com\" target=\"_blank\" rel=\"noopener\">www.the-numbers.com<\/a>.[\/caption]\r\n<h2>Find the domain of a function defined by an equation<\/h2>\r\nIn Functions and Function Notation, we were introduced to the concepts of <strong>domain and range<\/strong>. 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_0022.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/> <b>Figure 2<\/b>[\/caption]\r\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \"holding area\" that contains \"raw materials\" for a \"function machine\" and the range as another \"holding area\" for the machine\u2019s products.<span id=\"fs-id1165137737552\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137761714\">We can write the <strong>domain and range<\/strong> in <strong>interval notation<\/strong>, which uses values within brackets or parentheses to describe a set of numbers. In interval notation, we use a square bracket <strong>[<\/strong> when the set includes the endpoint and a parenthesis <strong>(<\/strong> 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>\r\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 even root, 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, exclude values that would make the radicand negative.<\/p>\r\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\r\n\r\n<ul id=\"fs-id1165135673417\">\r\n \t<li>The lowest term from the interval is written first.<\/li>\r\n \t<li>The greatest term in the interval is written second, following a comma.<\/li>\r\n \t<li>Parentheses, <strong>(<\/strong> or <strong>)<\/strong>, are used to signify that an endpoint is not included, called exclusive.<\/li>\r\n \t<li>Brackets, <strong>[<\/strong> or <strong>]<\/strong>, are used to indicate that an endpoint is included, called inclusive.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137807384\">The table below gives\u00a0a summary of interval notation.<span id=\"fs-id1165137406680\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/><\/span><\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\r\nFind the domain 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] .\r\n\r\n[reveal-answer q=\"425723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425723\"]\r\n\r\nFirst identify the input values. The input value is the first coordinate in an <strong>ordered pair<\/strong>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.\r\n<div style=\"text-align: center\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\r\n<p id=\"fs-id1165137466017\" style=\"text-align: center\">[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>\r\n[reveal-answer q=\"934906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"934906\"]\r\n\r\n[latex]\\left\\{-5,0,5,10,15\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134225655\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165134355557\"><strong>How To: Given a function written in equation form, find the domain.<\/strong><\/h3>\r\n<ol id=\"fs-id1165134187286\">\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\r\n \t<li>Write the domain in interval form, if possible.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Finding the Domain of a Function<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].\r\n\r\n[reveal-answer q=\"306869\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"306869\"]\r\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>\r\n<p id=\"fs-id1165135309759\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}[\/latex].\r\n\r\n[reveal-answer q=\"958539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"958539\"]\r\n\r\n[latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]164264[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137417188\" class=\"note precalculus howto textbox\">\r\n<h3>How To:\u00a0Given a function written in an equation form that includes a fraction, find the domain.<\/h3>\r\n<ol id=\"fs-id1165137463251\">\r\n \t<li>Identify the input values.<\/li>\r\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>\r\n \t<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Finding the Domain of a Function Involving a Denominator (Rational Function)<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].\r\n\r\n[reveal-answer q=\"677933\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"677933\"]\r\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>\r\n\r\n<div id=\"fs-id1165137736620\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{gathered}2-x=0 \\\\ -x=-2 \\\\ x=2 \\\\ \\text{ } \\end{gathered}[\/latex]<\/div>\r\nNow, 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]. 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].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010544\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/> <b>Figure 3<\/b>[\/caption]\r\n<p id=\"fs-id1165134036054\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function: [latex]f\\left(x\\right)=\\frac{1+4x}{2x - 1}[\/latex].\r\n\r\n[reveal-answer q=\"918828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"918828\"]\r\n\r\n[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]164323[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135527005\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137733733\">How To: Given a function written in equation form including an even root, find the domain.<\/h3>\r\n<ol id=\"fs-id1165137820030\">\r\n \t<li>Identify the input values.<\/li>\r\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>\r\n \t<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Finding the Domain of a Function with an Even Root<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].\r\n\r\n[reveal-answer q=\"360642\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"360642\"]\r\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>\r\n<p id=\"fs-id1165137749755\">Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137727831\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{gathered}7-x\\ge 0 \\\\ -x\\ge -7 \\\\ x\\le 7\\\\ \\text{ } \\end{gathered}[\/latex]<\/div>\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].\r\n\r\n[reveal-answer q=\"494621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"494621\"]\r\n\r\n[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]164263[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"\u201ctextbox\u201d textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can there be functions in which the domain and range do not intersect at all?<\/strong>\r\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>\r\n\r\n<\/div>\r\nIn 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 { }\u00a0are read as \"the set of,\" and the vertical bar | is read as \"such that,\" so we would read [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex] as \"the 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.\"\r\n<p id=\"fs-id1165135207589\">The table below compares inequality notation, set-builder notation, and interval notation.<\/p>\r\n\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Inequality Notation<\/th>\r\n<th>Set-builder Notation<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/1.png\"><img class=\" size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\r\n<td>5 &lt; <em>h<\/em>\u00a0\u2264 10<\/td>\r\n<td>{ <em>h<\/em> | 5 &lt; <em>h<\/em> \u2264 10}<\/td>\r\n<td>(5, 10]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/2.png\"><img class=\" size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\r\n<td>5 \u2264 <em>h<\/em> &lt; 10<\/td>\r\n<td>{ <em>h<\/em> | 5 \u2264 <em>h<\/em> &lt; 10}<\/td>\r\n<td>[5, 10]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/3.png\"><img class=\" size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\r\n<td>5 &lt; <em>h<\/em> &lt; 10<\/td>\r\n<td>{ <em>h<\/em> | 5 &lt; 10 }<\/td>\r\n<td>(5, 10)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/4.png\"><img class=\" size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\r\n<td><em>h<\/em> &lt; 10<\/td>\r\n<td>{ <em>h<\/em> | <em>h<\/em> &lt; 10 }<\/td>\r\n<td>( \u2212\u221e, 10)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/5.png\"><img class=\" size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\r\n<td><em>h<\/em> \u2265 10<\/td>\r\n<td>{ <em>h<\/em> | <em>h<\/em> \u2265 10 }<\/td>\r\n<td>[10,\u00a0\u221e )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/6.png\"><img class=\" size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\r\n<td>All real numbers<\/td>\r\n<td>\u211d<\/td>\r\n<td>(\u00a0\u2212\u221e,\u00a0\u221e )<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \"or.\" 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]\u00a0and [latex]\\left\\{4,6\\right\\}[\/latex]\u00a0is 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>\r\n\r\n<div id=\"fs-id1165135311695\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/div>\r\nThis video describes how to use interval notation to describe a set.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=hqg85P0ZMZ4\r\n\r\nThis video describes how to use Set-Builder notation to describe a set.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=rPcGeaDRnyc&amp;feature=youtu.be\r\n<div class=\"title textbox\">\r\n\r\n&nbsp;\r\n<h3>A General Note: Set-Builder Notation and Interval Notation<\/h3>\r\nSet-builder notation 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, \"the set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.\" For example,\r\n<p style=\"text-align: center\">[latex]\\left\\{x|4&lt;x\\le 12\\right\\}[\/latex]<\/p>\r\n<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,\r\n<p style=\"text-align: center\">[latex]\\left(4,12\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137805770\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137423878\">How To: Given a line graph, describe the set of values using interval notation.<\/h3>\r\n<ol id=\"fs-id1165134032280\">\r\n \t<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\r\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>\r\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>\r\n \t<li>Use the union symbol [latex]\\cup [\/latex] to combine all intervals into one set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_02_05\" class=\"example\">\r\n<div id=\"fs-id1165134342702\" class=\"exercise\">\r\n<div id=\"fs-id1165137803670\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Describing Sets on the Real-Number Line<\/h3>\r\nDescribe the intervals of values shown in Figure 4\u00a0using inequality notation, set-builder notation, and interval notation.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"865588\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"865588\"]\r\n<p id=\"fs-id1165135412905\">To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, \" [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.\"<\/p>\r\n\r\n<table id=\"fs-id1165137447518\" class=\"unnumbered\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Inequality<\/strong><\/td>\r\n<td>[latex]1\\le x\\le 3\\text{or}x&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Set-builder notation<\/strong><\/td>\r\n<td>[latex]\\left\\{x|1\\le x\\le 3\\text{or}x&gt;5\\right\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Interval notation<\/strong><\/td>\r\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135341412\">Given Figure 5, specify the graphed set in<\/p>\r\n\r\n<ol id=\"fs-id1165137595582\">\r\n \t<li>words<\/li>\r\n \t<li>set-builder notation<\/li>\r\n \t<li>interval notation<\/li>\r\n<\/ol>\r\n<figure id=\"Figure_01_02_005\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0052.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/> <b>Figure 5<\/b>[\/caption]<\/figure>\r\n[reveal-answer q=\"426537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"426537\"]\r\n<ol>\r\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>\r\n \t<li>[latex]\\left\\{x|x\\le -2\\text{or}-1\\le x&lt;3\\right\\}[\/latex]<\/li>\r\n \t<li>[latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nFinding Domain and Range from Graphs\r\n\r\nAnother 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 Figure 6.<span id=\"fs-id1165137432156\">\r\n<\/span>\r\n<div class=\"\u201ctextbox\u201d\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0062.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\" \/> <b>Figure 6<\/b>[\/caption]\r\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>\r\n\r\n<div id=\"Example_01_02_06\" class=\"example\">\r\n<div id=\"fs-id1165137561401\" class=\"exercise\">\r\n<div id=\"fs-id1165137599824\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Finding Domain and Range from a Graph<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex]\u00a0whose graph is shown in Figure 7.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"916064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"916064\"]\r\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]\u00a0is [latex]\\left(-3,1\\right][\/latex].<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/> <b>Figure 8<\/b>[\/caption]\r\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].<\/p>\r\n[\/hidden-answer]<b><\/b><span id=\"fs-id1165137937577\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it 9<\/h3>\r\n[ohm_question hide_question_numbers=1]30605[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=QAxZEelInJc\r\n<div id=\"Example_01_02_07\" class=\"example\">\r\n<div id=\"fs-id1165134182686\" class=\"exercise\">\r\n<div id=\"fs-id1165137461643\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Finding Domain and Range from a Graph of Oil Production<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex] whose graph is shown in Figure 9.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"489\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0092.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\" \/> <b>Figure 9.<\/b> (credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)[\/caption]\r\n\r\n[reveal-answer q=\"579613\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"579613\"]\r\n<p id=\"fs-id1165137476085\">The input quantity along the horizontal axis is \"years,\" which we represent with the variable [latex]t[\/latex] for time. The output quantity is \"thousands of barrels of oil per day,\" 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>\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGiven the graph in Figure 10, identify the domain and range using interval notation.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0102.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\" \/> <b>Figure 10<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"420935\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"420935\"]\r\n\r\nDomain = [1950, 2002] \u00a0 Range = [47,000,000, 89,000,000]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137434590\" class=\"note precalculus qa textbox\">\r\n<h3 id=\"fs-id1165137812796\">Q &amp; A<\/h3>\r\n<strong>Can a function\u2019s domain and range be the same?<\/strong>\r\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>\r\n\r\n<\/div>\r\n<span id=\"fs-id1165137432156\">Finding Domain and Range from Graphs<\/span>\r\n<div class=\"\u201ctextbox\u201d\">\r\n\r\nWe will now return to our set of toolkit functions to determine the domain and range of each.\r\n\r\n<\/div>\r\n<section id=\"fs-id1165134384565\">\r\n<figure id=\"Figure_01_02_011\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0112.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/> 11[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 11.<\/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].<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_012\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0122.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/> 12[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 12.<\/strong> For the <strong>identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_013\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0132.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/> 13[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 13.<\/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.<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_014\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0142.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/> 14[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 14.<\/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.<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_015\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0152.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/> 15[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 15.<\/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.<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_016\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/> 16[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 16.<\/strong> For the reciprocal function [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.<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_017\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/> 17[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 17.<\/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.<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_018\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/> 18[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 18.<\/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].<\/p>\r\n<\/figure>\r\n<figure id=\"Figure_01_02_019\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/> 19[\/caption]\r\n<p style=\"text-align: center\"><strong>Figure 19.<\/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).<\/p>\r\n<\/figure>\r\n<div id=\"fs-id1165137462732\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137611181\">How To: Given the formula for a function, determine the domain and range.<\/h3>\r\n<ol id=\"fs-id1165137405229\">\r\n \t<li>Exclude from the domain any input values that result in division by zero.<\/li>\r\n \t<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\r\n \t<li>Use the valid input values to determine the range of the output values.<\/li>\r\n \t<li>Look at the function graph and table values to confirm the actual function behavior.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_02_08\" class=\"example\">\r\n<div id=\"fs-id1165137558723\" class=\"exercise\">\r\n<div id=\"fs-id1165137464274\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Finding the Domain and Range Using Toolkit Functions<\/h3>\r\n<p id=\"fs-id1165135613224\">Find the domain and range of [latex]f\\left(x\\right)=2{x}^{3}-x[\/latex].<\/p>\r\n[reveal-answer q=\"86473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"86473\"]\r\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>\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_02_09\" class=\"example\">\r\n<div id=\"fs-id1165137448155\" class=\"exercise\">\r\n<div id=\"fs-id1165137661316\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Finding the Domain and Range<\/h3>\r\n<p id=\"fs-id1165137419507\">Find the domain and range of [latex]f\\left(x\\right)=\\frac{2}{x+1}[\/latex].<\/p>\r\n[reveal-answer q=\"962522\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"962522\"]\r\n\r\nWe 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 10: Finding the Domain and Range<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].\r\n\r\n[reveal-answer q=\"792421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792421\"]\r\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>\r\n\r\n<div id=\"eip-id1165137567088\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/div>\r\n<p id=\"fs-id1165137465335\">The domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].<\/p>\r\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].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nFigure 20\u00a0represents the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0202.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/> <b>Figure 20<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=-\\sqrt{2-x}[\/latex].\r\n\r\n[reveal-answer q=\"298242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"298242\"]\r\n\r\nDomain: [latex]\\left(-\\infty ,2\\right][\/latex] \u00a0 Range: [latex]\\left(-\\infty ,0\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<div class=\"\u201ctextbox\u201d\">\r\n<h2 class=\"mceTemp\">Graphic Piecewise-Defined Functions<\/h2>\r\nSometimes, 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, <strong>absolute value<\/strong> can be defined as the <strong>magnitude<\/strong>, or <strong>modulus<\/strong>, 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.\r\n<p id=\"fs-id1165137558775\">If we input 0, or a positive value, the output is the same as the input.<\/p>\r\n\r\n<div id=\"fs-id1165135194329\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]f\\left(x\\right)=x\\text{ if }x\\ge 0[\/latex]<\/div>\r\n<p id=\"fs-id1165137529947\">If we input a negative value, the output is the opposite of the input.<\/p>\r\n\r\n<div id=\"fs-id1165133112779\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]f\\left(x\\right)=-x\\text{ if }x&lt;0[\/latex]<\/div>\r\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 <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\r\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 \"boundaries.\" 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, S, would be\u00a00.1S if [latex]{S}\\le\\[\/latex] $10,000\u00a0and 1000 + 0.2 (S - $10,000),\u00a0if S&gt; $10,000.<\/p>\r\n\r\n<div id=\"fs-id1165137531241\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Piecewise Function<\/h3>\r\n<p id=\"fs-id1165135504970\">A piecewise function 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>\r\n<p style=\"text-align: center\">[latex] f\\left(x\\right)=\\begin{cases}\\text{formula 1 if x is in domain 1}\\\\ \\text{formula 2 if x is in domain 2}\\\\ \\text{formula 3 if x is in domain 3}\\end{cases} [\/latex]<\/p>\r\nIn piecewise notation, the absolute value function is\r\n<p style=\"text-align: center\">[latex]|x|=\\begin{cases}\\begin{align}&amp;x&amp;&amp;\\text{ if }x\\ge 0\\\\ &amp;-x&amp;&amp;\\text{ if }x&lt;0\\end{align}\\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137768426\" class=\"note precalculus howto textbox\">\r\n<h3>How To:\u00a0Given a piecewise function, write the formula and identify the domain for each interval.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165135443772\">\r\n \t<li>Identify the intervals for which different rules apply.<\/li>\r\n \t<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\r\n \t<li>Use braces and if-statements to write the function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 11: Writing a Piecewise Function<\/h3>\r\nA 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 <strong>function<\/strong> relating the number of people, [latex]n[\/latex], to the cost, [latex]C[\/latex].\r\n\r\n[reveal-answer q=\"525510\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525510\"]\r\n\r\nTwo different formulas will be needed. For <em>n<\/em>-values under 10, C=5n. For values of n that are 10 or greater, C=50.\r\n<p style=\"text-align: center\">[latex]C(n)=\\begin{cases}\\begin{align}{5n}&amp;\\hspace{5mm}\\text{ if }{0}&lt;{n}&lt;{10}\\\\ 50&amp;\\hspace{5mm}\\text{ if }{n}\\ge 10\\end{align}\\end{cases}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe function is represented in Figure 21. 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"360\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0212.jpg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/> <b>Figure 21<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=B1jfpiI-QQ8&amp;feature=youtu.be\r\n<div id=\"Example_01_02_12\" class=\"example\">\r\n<div id=\"fs-id1165135436662\" class=\"exercise\">\r\n<div id=\"fs-id1165135436664\" class=\"problem textbox shaded\">\r\n<h3>Example 12: Working with a Piecewise Function<\/h3>\r\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>\r\n\r\n<div id=\"fs-id1165137660470\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]C\\left(g\\right)=\\begin{cases}\\begin{align}&amp;{25} &amp;&amp;\\hspace{-5mm}\\text{ if }{ 0 }&lt;{ g }&lt;{ 2 }\\\\ &amp;{ 25+10 }\\left(g - 2\\right) &amp;&amp;\\hspace{-5mm}\\text{ if }{ g}\\ge{ 2 }\\end{align}\\end{cases}[\/latex]<\/div>\r\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>\r\n[reveal-answer q=\"67822\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"67822\"]\r\n<p id=\"fs-id1165134373545\">To find the cost of using 1.5 gigabytes of data, C(1.5), 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>\r\n<p style=\"text-align: center\">[latex]C(1.5) = \\$25[\/latex]<\/p>\r\n<p id=\"fs-id1165135440213\">To find the cost of using 4 gigabytes of data, C(4), we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\r\n\r\n<div style=\"text-align: center\">[latex]C(4)=25 + 10( 4-2) =\\$45[\/latex]<\/div>\r\n<h4>Analysis of the Solution<\/h4>\r\nThe function is represented in Figure 22. 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0222.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/> <b>Figure 22<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137600493\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135532516\">How To:\u00a0Given a piecewise function, sketch a graph.<\/h3>\r\n<ol id=\"fs-id1165137588539\">\r\n \t<li>Indicate on the <em>x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\r\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>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_02_13\" class=\"example\">\r\n<div id=\"fs-id1165137781618\" class=\"exercise\">\r\n<div id=\"fs-id1165135412870\" class=\"problem textbox shaded\">\r\n<h3>Example 13: Graphing a Piecewise Function<\/h3>\r\n<p id=\"fs-id1165137838785\">Sketch a graph of the function.<\/p>\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}&amp;{ x }^{2} &amp;&amp;\\hspace{-5mm}\\text{ if }{ x }\\le{ 1 }\\\\ &amp;{ 3 } &amp;&amp;\\hspace{-5mm}\\text{ if } { 1 }&amp;lt{ x }\\le 2\\\\ &amp;{ x } &amp;&amp;\\hspace{-5mm}\\text{ if }{ x }&amp;gt{ 2 }\\end{align}\\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"617292\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617292\"]\r\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>\r\n<p id=\"fs-id1165137642848\">Below are\u00a0the three components of the piecewise function graphed on separate coordinate systems.<\/p>\r\n\r\n<figure id=\"Figure_01_02_023\"><figcaption>(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>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"974\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010549\/CNX_Precalc_Figure_01_02_023abc2.jpg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/> <b>Figure 23<\/b>[\/caption]<\/figure>\r\n<p id=\"fs-id1165137676209\">Now that we have sketched each piece individually, we combine them in the same coordinate plane.<span id=\"fs-id1165137646696\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010549\/CNX_Precalc_Figure_01_02_0262.jpg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/> <b>Figure 24<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\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]\u00a0and [latex]\\left(2,3\\right)[\/latex] are.<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGraph the following piecewise function.\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=\\begin{cases}{ x}^{3} \\text{ if }{ x }&amp;lt{-1 }\\\\ { -2 } \\text{ if } { -1 }&amp;lt{ x }&amp;lt{ 4 }\\\\ \\sqrt{x} \\text{ if }{ x }&amp;gt{ 4 }\\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"836663\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"836663\"]\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010549\/CNX_Precalc_Figure_01_02_0272.jpg\" alt=\"Graph of f(x).\" width=\"487\" height=\"408\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]32883[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137810682\" class=\"note precalculus qa textbox\">\r\n<h3>Q&amp;A<\/h3>\r\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>\r\n<p id=\"fs-id1165137464467\"><em>No. Each value corresponds to one equation in a piecewise formula.<\/em><\/p>\r\n\r\n<\/div>\r\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Key Concepts<\/span><\/h2>\r\n<ul id=\"fs-id1165137591772\">\r\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>\r\n \t<li>The domain of a function can be determined by listing the input values of a set of ordered pairs.<\/li>\r\n \t<li>The domain of a function can also be determined by identifying the input values of a function written as an equation.<\/li>\r\n \t<li>Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.<\/li>\r\n \t<li>For many functions, the domain and range can be determined from a graph.<\/li>\r\n \t<li>An understanding of toolkit functions can be used to find the domain and range of related functions.<\/li>\r\n \t<li>A piecewise function is described by more than one formula.<\/li>\r\n \t<li>A piecewise function can be graphed using each algebraic formula on its assigned subdomain.<\/li>\r\n<\/ul>\r\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Glossary<\/span><\/h2>\r\n<dl id=\"fs-id1165135445751\" class=\"definition\">\r\n \t<dt><strong>interval notation<\/strong><\/dt>\r\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>\r\n<\/dl>\r\n<dl id=\"fs-id1165135487256\" class=\"definition\">\r\n \t<dt><strong>piecewise function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137452169\">a function in which more than one formula is used to define the output<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137863188\" class=\"definition\">\r\n \t<dt><strong>set-builder notation<\/strong><\/dt>\r\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>\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Section 1.6 Homework Exercises<\/span><\/h2>\r\n1. Why does the domain differ for different functions?\r\n\r\n2. How do we determine the domain of a function defined by an equation?\r\n\r\n3. 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].\r\n\r\n4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?\r\n\r\n5. How do you graph a piecewise function?\r\n<p id=\"fs-id1165137408926\">For the following exercises, find the domain of each function using interval notation.<\/p>\r\n6. [latex]f\\left(x\\right)=-2x\\left(x - 1\\right)\\left(x - 2\\right)[\/latex]\r\n\r\n7. [latex]f\\left(x\\right)=5 - 2{x}^{2}[\/latex]\r\n\r\n8. [latex]f\\left(x\\right)=3\\sqrt{x - 2}[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=3-\\sqrt{6 - 2x}[\/latex]\r\n\r\n10. [latex]f\\left(x\\right)=\\sqrt{4 - 3x}[\/latex]\r\n\r\n11. [latex]f\\left(x\\right)=\\sqrt{{x}^{2}+4}[\/latex]\r\n\r\n12. [latex]f\\left(x\\right)=\\sqrt[3]{1 - 2x}[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex]\r\n\r\n14. [latex]f\\left(x\\right)=\\frac{9}{x - 6}[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=\\frac{3x+1}{4x+2} [\/latex]\r\n\r\n16. [latex]f\\left(x\\right)=\\frac{\\sqrt{x+4}}{x - 4} [\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=\\frac{x - 3}{{x}^{2}+9x - 22} [\/latex]\r\n\r\n18. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-x - 6} [\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=\\frac{2{x}^{3}-250}{{x}^{2}-2x - 15} [\/latex]\r\n\r\n20. [latex]\\frac{5}{\\sqrt{x - 3}} [\/latex]\r\n\r\n21. [latex]\\frac{2x+1}{\\sqrt{5-x}} [\/latex]\r\n\r\n22. [latex]f\\left(x\\right)=\\frac{\\sqrt{x - 4}}{\\sqrt{x - 6}} [\/latex]\r\n\r\n23. [latex]f\\left(x\\right)=\\frac{\\sqrt{x - 6}}{\\sqrt{x - 4}} [\/latex]\r\n\r\n24. [latex]f\\left(x\\right)=\\frac{x}{x} [\/latex]\r\n\r\n25. [latex]f\\left(x\\right)=\\frac{{x}^{2}-9x}{{x}^{2}-81} [\/latex]\r\n\r\n26. Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{2{x}^{3}-50x} [\/latex] by:\r\n<div style=\"margin: 0 0 0 40px;border: none;padding: 0px\">a. using algebra.\r\nb. graphing the function in the radicand and determining intervals on the <em>x<\/em>-axis for which the radicand is nonnegative.<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\nFor the following exercises, write the domain and range of each function using interval notation.\r\n\r\n27.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005038\/CNX_Precalc_Figure_01_02_202.jpg\" alt=\"Graph of a function from (2, 8].\" width=\"487\" height=\"222\" \/>\r\n\r\nDomain: ________ Range: ________\r\n\r\n28.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005038\/CNX_Precalc_Figure_01_02_203.jpg\" alt=\"Graph of a function from [4, 8).\" width=\"487\" height=\"222\" \/>\r\n\r\n29.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005038\/CNX_Precalc_Figure_01_02_204.jpg\" alt=\"Graph of a function from [-4, 4].\" width=\"487\" height=\"220\" \/>\r\n\r\n30.\r\n\r\n&nbsp;\r\n<div id=\"fs-id1165135245908\" class=\"problem\">\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_205.jpg\" alt=\"Graph of a function from [2, 6].\" width=\"487\" height=\"282\" \/>\r\n\r\n31.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_206.jpg\" alt=\"Graph of a function from [-5, 3).\" width=\"487\" height=\"189\" \/>\r\n\r\n32.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_207.jpg\" alt=\"Graph of a function from [-3, 2).\" width=\"487\" height=\"377\" \/>\r\n\r\n33.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_208.jpg\" alt=\"Graph of a function from (-infinity, 2].\" width=\"487\" height=\"220\" \/>\r\n\r\n34.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_209.jpg\" alt=\"Graph of a function from [-4, infinity).\" width=\"487\" height=\"316\" \/>\r\n\r\n35.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_210.jpg\" alt=\"Graph of a function from [-6, -1\/6]U[1\/6, 6]\/.\" width=\"975\" height=\"442\" \/>\r\n\r\n36.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_211.jpg\" alt=\"Graph of a function from (-2.5, infinity).\" width=\"487\" height=\"535\" \/>\r\n\r\n37.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_212.jpg\" alt=\"Graph of a function from [-3, infinity).\" width=\"975\" height=\"379\" \/>\r\n\r\nFor the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.\r\n\r\n38. [latex]f(x)=\\begin{cases}{x}+{1}&amp;\\text{ if }&amp;{ x }&lt;{ -2 } \\\\{-2x - 3}&amp;\\text{ if }&amp;{ x }\\ge { -2 }\\\\ \\end{cases} [\/latex]\r\n\r\n39. [latex]f\\left(x\\right)=\\begin{cases}{2x - 1}&amp;\\text{ if }&amp;{ x }&lt;{ 1 }\\\\ {1+x }&amp;\\text{ if }&amp;{ x }\\ge{ 1 } \\end{cases}[\/latex]\r\n\r\n40. [latex]f\\left(x\\right)=\\begin{cases}{x+1}&amp;\\text{ if }&amp;{ x }&lt;{ 0 }\\\\ {x - 1 }&amp;\\text{ if }&amp;{ x }&gt;{ 0 }\\end{cases}[\/latex]\r\n\r\n41. [latex]f\\left(x\\right)=\\begin{cases}{3} &amp;\\text{ if }&amp;{ x } &lt;{ 0 }\\\\ \\sqrt{x}&amp;\\text{ if }&amp;{ x }\\ge { 0 }\\end{cases}[\/latex]\r\n\r\n42. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}&amp;\\text{ if }&amp;{ x } &lt;{ 0 }\\\\ {1-x}&amp;\\text{ if }&amp;{ x } &gt;{ 0 }\\end{cases}[\/latex]\r\n\r\n43. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}&amp;\\text{ if }&amp;{ x }&lt;{ 0 }\\\\ {x+2 }&amp;\\text{ if }&amp;{ x }\\ge { 0 }\\end{cases}[\/latex]\r\n\r\n44. [latex]f\\left(x\\right)=\\begin{cases}x+1&amp; \\text{if}&amp; x&lt;1\\\\ {x}^{3}&amp; \\text{if}&amp; x\\ge 1\\end{cases}[\/latex]\r\n\r\n45. [latex]f\\left(x\\right)=\\begin{cases}|x|&amp;\\text{ if }&amp;{ x }&lt;{ 2 }\\\\ { 1 }&amp;\\text{ if }&amp;{ x }\\ge{ 2 }\\end{cases}[\/latex]\r\nFor 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].\r\n\r\n46. [latex]f\\left(x\\right)=\\begin{cases}{ x+1 }&amp;\\text{ if }&amp;{ x }&lt;{ -2 }\\\\ { -2x - 3 }&amp;\\text{ if }&amp;{ x }\\ge{ -2 }\\end{cases}[\/latex]\r\n\r\n47. [latex]f\\left(x\\right)=\\begin{cases}{ 1 }&amp;\\text{ if }&amp;{ x }\\le{ -3 }\\\\{ 0 }&amp;\\text{ if }&amp;{ x }&gt;{ -3 }\\end{cases}[\/latex]\r\n\r\n48. [latex]f\\left(x\\right)=\\begin{cases}{-2}{x}^{2}+{ 3 }&amp;\\text{ if }&amp;{ x }\\le { -1 }\\\\ { 5x } - { 7 } &amp;\\text{ if }&amp;{ x } &gt; { -1 }\\end{cases}[\/latex]\r\nFor 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].\r\n\r\n49. [latex]f\\left(x\\right)=\\begin{cases}{ 7x+3 }&amp;\\text{ if }&amp;{ x }&lt;{ 0 }\\\\{ 7x+6 }&amp;\\text{ if }&amp;{ x }\\ge{ 0 }\\end{cases}[\/latex]\r\n\r\n50. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}{ -2 }&amp;\\text{ if }&amp;{ x }&lt;{ 2 }\\\\{ 4+|x - 5|}&amp;\\text{ if }&amp;{ x }\\ge{ 2 }\\end{cases}[\/latex]\r\n\r\n51. [latex]f\\left(x\\right)=\\begin{cases}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{cases}[\/latex]\r\nFor the following exercises, write the domain for the piecewise function in interval notation.\r\n\r\n52. [latex]f\\left(x\\right)=\\begin{cases}{x+1}&amp;\\text{ if }&amp;{ x }&lt;{ -2 }\\\\{ -2x - 3}&amp;\\text{ if }&amp;{ x }\\ge{ -2 }\\end{cases}[\/latex]\r\n\r\n53. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}{ -2 }&amp;\\text{ if}&amp;{ x }&lt;{ 1 }\\\\{-x}^{2}+{2}&amp;\\text{ if }&amp;{ x }&gt;{ 1 }\\end{cases}[\/latex]\r\n\r\n54. [latex]f\\left(x\\right)=\\begin{cases}{ 2x - 3 }&amp;\\text{ if }&amp;{ x }&lt;{ 0 }\\\\{ -3}{x}^{2}&amp;\\text{ if }&amp;{ x }\\ge{ 2 }\\end{cases}[\/latex]\r\n\r\n55. 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.\r\n\r\n56. 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.\r\n\r\n57. 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]\r\n\r\n58. Create a function in which the range is all nonnegative real numbers.\r\n\r\n59 .Create a function in which the domain is [latex]x&gt;2[\/latex].\r\n\r\n60. The cost in dollars of making [latex]x[\/latex] items is given by the function [latex]C\\left(x\\right)=10x+500[\/latex].\r\n<div style=\"margin: 0 0 0 40px;border: none;padding: 0px\">A. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.\r\nB. What is the cost of making 25 items?\r\nC. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, [latex]C\\left(x\\right)?[\/latex]<\/div>\r\n61. 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]. What is the domain of the function? What does the domain mean in the context of the problem?\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the domain of a function defined by an equation.<\/li>\n<li>Find the domain of a function from its graph.<\/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>. Figure 1\u00a0shows 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 <strong>domain<\/strong> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_0012.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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> Based on data compiled by <a href=\"http:\/\/www.the-numbers.com\" target=\"_blank\" rel=\"noopener\">www.the-numbers.com<\/a>.<\/p>\n<\/div>\n<h2>Find the domain of a function defined by an equation<\/h2>\n<p>In Functions and Function Notation, we were introduced to the concepts of <strong>domain and range<\/strong>. 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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_0022.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135453892\">We can visualize the domain as a &#8220;holding area&#8221; that contains &#8220;raw materials&#8221; for a &#8220;function machine&#8221; and the range as another &#8220;holding area&#8221; for the machine\u2019s products.<span id=\"fs-id1165137737552\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137761714\">We can write the <strong>domain and range<\/strong> in <strong>interval notation<\/strong>, which uses values within brackets or parentheses to describe a set of numbers. In interval notation, we use a square bracket <strong>[<\/strong> when the set includes the endpoint and a parenthesis <strong>(<\/strong> 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 even root, 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, exclude 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 lowest term from the interval is written first.<\/li>\n<li>The greatest term in the interval is written second, following a comma.<\/li>\n<li>Parentheses, <strong>(<\/strong> or <strong>)<\/strong>, are used to signify that an endpoint is not included, called exclusive.<\/li>\n<li>Brackets, <strong>[<\/strong> or <strong>]<\/strong>, are used to indicate that an endpoint is included, called inclusive.<\/li>\n<\/ul>\n<p id=\"fs-id1165137807384\">The table below gives\u00a0a summary of interval notation.<span id=\"fs-id1165137406680\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/><\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\n<p>Find the domain 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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q425723\">Show Solution<\/span><\/p>\n<div id=\"q425723\" class=\"hidden-answer\" style=\"display: none\">\n<p>First identify the input values. The input value is the first coordinate in an <strong>ordered pair<\/strong>. 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 style=\"text-align: center\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\n<p id=\"fs-id1165137466017\" style=\"text-align: center\">[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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q934906\">Show Solution<\/span><\/p>\n<div id=\"q934906\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{-5,0,5,10,15\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134225655\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165134355557\"><strong>How To: Given a function written in equation form, find the domain.<\/strong><\/h3>\n<ol id=\"fs-id1165134187286\">\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 shaded\">\n<h3>Example 2: Finding the Domain of a Function<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q306869\">Show Solution<\/span><\/p>\n<div id=\"q306869\" class=\"hidden-answer\" style=\"display: none\">\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<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q958539\">Show Solution<\/span><\/p>\n<div id=\"q958539\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm164264\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=164264&theme=oea&iframe_resize_id=ohm164264\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165137417188\" class=\"note precalculus howto textbox\">\n<h3>How To:\u00a0Given a function written in an equation form that includes a fraction, find the domain.<\/h3>\n<ol id=\"fs-id1165137463251\">\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 shaded\">\n<h3>Example 3: Finding the Domain of a Function Involving a Denominator (Rational Function)<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q677933\">Show Solution<\/span><\/p>\n<div id=\"q677933\" class=\"hidden-answer\" style=\"display: none\">\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=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{gathered}2-x=0 \\\\ -x=-2 \\\\ x=2 \\\\ \\text{ } \\end{gathered}[\/latex]<\/div>\n<p>Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[\/latex] or [latex]x>2[\/latex]. 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 style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010544\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134036054\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  The Domain of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function: [latex]f\\left(x\\right)=\\frac{1+4x}{2x - 1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q918828\">Show Solution<\/span><\/p>\n<div id=\"q918828\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm164323\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=164323&theme=oea&iframe_resize_id=ohm164323\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165135527005\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137733733\">How To: Given a function written in equation form including an even root, find the domain.<\/h3>\n<ol id=\"fs-id1165137820030\">\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 shaded\">\n<h3>Example 4: Finding the Domain of a Function with an Even Root<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q360642\">Show Solution<\/span><\/p>\n<div id=\"q360642\" class=\"hidden-answer\" style=\"display: none\">\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=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{gathered}7-x\\ge 0 \\\\ -x\\ge -7 \\\\ x\\le 7\\\\ \\text{ } \\end{gathered}[\/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<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Domain and Range of Square Root Functions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q494621\">Show Solution<\/span><\/p>\n<div id=\"q494621\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm164263\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=164263&theme=oea&iframe_resize_id=ohm164263\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"\u201ctextbox\u201d textbox\">\n<h3>Q &amp; A<\/h3>\n<p><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<p>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 { }\u00a0are read as &#8220;the set of,&#8221; and the vertical bar | is read as &#8220;such that,&#8221; so we would read [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex] as &#8220;the 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.&#8221;<\/p>\n<p id=\"fs-id1165135207589\">The table below compares inequality notation, set-builder notation, and interval notation.<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th>Inequality Notation<\/th>\n<th>Set-builder Notation<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\n<td>5 &lt; <em>h<\/em>\u00a0\u2264 10<\/td>\n<td>{ <em>h<\/em> | 5 &lt; <em>h<\/em> \u2264 10}<\/td>\n<td>(5, 10]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012924\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\n<td>5 \u2264 <em>h<\/em> &lt; 10<\/td>\n<td>{ <em>h<\/em> | 5 \u2264 <em>h<\/em> &lt; 10}<\/td>\n<td>[5, 10]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\n<td>5 &lt; <em>h<\/em> &lt; 10<\/td>\n<td>{ <em>h<\/em> | 5 &lt; 10 }<\/td>\n<td>(5, 10)<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\n<td><em>h<\/em> &lt; 10<\/td>\n<td>{ <em>h<\/em> | <em>h<\/em> &lt; 10 }<\/td>\n<td>( \u2212\u221e, 10)<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\n<td><em>h<\/em> \u2265 10<\/td>\n<td>{ <em>h<\/em> | <em>h<\/em> \u2265 10 }<\/td>\n<td>[10,\u00a0\u221e )<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012925\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\n<td>All real numbers<\/td>\n<td>\u211d<\/td>\n<td>(\u00a0\u2212\u221e,\u00a0\u221e )<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word &#8220;or.&#8221; 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]\u00a0and [latex]\\left\\{4,6\\right\\}[\/latex]\u00a0is 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=\"equation unnumbered\" style=\"text-align: center\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/div>\n<p>This video describes how to use interval notation to describe a set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hqg85P0ZMZ4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This video describes how to use Set-Builder notation to describe a set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Set-Builder Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rPcGeaDRnyc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"title textbox\">\n<p>&nbsp;<\/p>\n<h3>A General Note: Set-Builder Notation and Interval Notation<\/h3>\n<p>Set-builder notation 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, &#8220;the set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.&#8221; For example,<\/p>\n<p style=\"text-align: center\">[latex]\\left\\{x|4<x\\le 12\\right\\}[\/latex]<\/p>\n<p><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<p style=\"text-align: center\">[latex]\\left(4,12\\right][\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137805770\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137423878\">How To: Given a line graph, describe the set of values using interval notation.<\/h3>\n<ol id=\"fs-id1165134032280\">\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 id=\"Example_01_02_05\" class=\"example\">\n<div id=\"fs-id1165134342702\" class=\"exercise\">\n<div id=\"fs-id1165137803670\" class=\"problem textbox shaded\">\n<h3>Example 5: Describing Sets on the Real-Number Line<\/h3>\n<p>Describe the intervals of values shown in Figure 4\u00a0using inequality notation, set-builder notation, and interval notation.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q865588\">Show Solution<\/span><\/p>\n<div id=\"q865588\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135412905\">To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, &#8221; [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.&#8221;<\/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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135341412\">Given Figure 5, specify the graphed set in<\/p>\n<ol id=\"fs-id1165137595582\">\n<li>words<\/li>\n<li>set-builder notation<\/li>\n<li>interval notation<\/li>\n<\/ol>\n<figure id=\"Figure_01_02_005\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0052.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q426537\">Show Solution<\/span><\/p>\n<div id=\"q426537\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\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]<\/li>\n<li>[latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Finding Domain and Range from Graphs<\/p>\n<p>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 Figure 6.<span id=\"fs-id1165137432156\"><br \/>\n<\/span><\/p>\n<div class=\"\u201ctextbox\u201d\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0062.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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\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 id=\"Example_01_02_06\" class=\"example\">\n<div id=\"fs-id1165137561401\" class=\"exercise\">\n<div id=\"fs-id1165137599824\" class=\"problem textbox shaded\">\n<h3>Example 6: Finding Domain and Range from a Graph<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex]\u00a0whose graph is shown in Figure 7.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q916064\">Show Solution<\/span><\/p>\n<div id=\"q916064\" class=\"hidden-answer\" style=\"display: none\">\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]\u00a0is [latex]\\left(-3,1\\right][\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010545\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\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].<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><span id=\"fs-id1165137937577\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it 9<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm30605\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=30605&theme=oea&iframe_resize_id=ohm30605\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 1:  Determine the Domain and Range of the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"Example_01_02_07\" class=\"example\">\n<div id=\"fs-id1165134182686\" class=\"exercise\">\n<div id=\"fs-id1165137461643\" class=\"problem textbox shaded\">\n<h3>Example 7: Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex] whose graph is shown in Figure 9.<\/p>\n<div style=\"width: 499px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0092.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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9.<\/b> (credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q579613\">Show Solution<\/span><\/p>\n<div id=\"q579613\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137476085\">The input quantity along the horizontal axis is &#8220;years,&#8221; which we represent with the variable [latex]t[\/latex] for time. The output quantity is &#8220;thousands of barrels of oil per day,&#8221; 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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Given the graph in Figure 10, identify the domain and range using interval notation.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0102.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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q420935\">Show Solution<\/span><\/p>\n<div id=\"q420935\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [1950, 2002] \u00a0 Range = [47,000,000, 89,000,000]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434590\" class=\"note precalculus qa textbox\">\n<h3 id=\"fs-id1165137812796\">Q &amp; A<\/h3>\n<p><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<p><span id=\"fs-id1165137432156\">Finding Domain and Range from Graphs<\/span><\/p>\n<div class=\"\u201ctextbox\u201d\">\n<p>We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n<\/div>\n<section id=\"fs-id1165134384565\">\n<figure id=\"Figure_01_02_011\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0112.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">11<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 11.<\/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].<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_012\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010546\/CNX_Precalc_Figure_01_02_0122.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">12<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 12.<\/strong> For the <strong>identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_013\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0132.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">13<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 13.<\/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.<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_014\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0142.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">14<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 14.<\/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.<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_015\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0152.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/><\/p>\n<p class=\"wp-caption-text\">15<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 15.<\/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.<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_016\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">16<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 16.<\/strong> For the reciprocal function [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.<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_017\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010547\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">17<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 17.<\/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.<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_018\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">18<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 18.<\/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].<\/p>\n<\/figure>\n<figure id=\"Figure_01_02_019\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">19<\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Figure 19.<\/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).<\/p>\n<\/figure>\n<div id=\"fs-id1165137462732\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137611181\">How To: Given the formula for a function, determine the domain and range.<\/h3>\n<ol id=\"fs-id1165137405229\">\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 id=\"Example_01_02_08\" class=\"example\">\n<div id=\"fs-id1165137558723\" class=\"exercise\">\n<div id=\"fs-id1165137464274\" class=\"problem textbox shaded\">\n<h3>Example 8: 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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q86473\">Show Solution<\/span><\/p>\n<div id=\"q86473\" class=\"hidden-answer\" style=\"display: none\">\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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_02_09\" class=\"example\">\n<div id=\"fs-id1165137448155\" class=\"exercise\">\n<div id=\"fs-id1165137661316\" class=\"problem textbox shaded\">\n<h3>Example 9: 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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q962522\">Show Solution<\/span><\/p>\n<div id=\"q962522\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 10: Finding the Domain and Range<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792421\">Show Solution<\/span><\/p>\n<div id=\"q792421\" class=\"hidden-answer\" style=\"display: none\">\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=\"equation unnumbered\" style=\"text-align: center\">[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].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Figure 20\u00a0represents the function [latex]f[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0202.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=-\\sqrt{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q298242\">Show Solution<\/span><\/p>\n<div id=\"q298242\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain: [latex]\\left(-\\infty ,2\\right][\/latex] \u00a0 Range: [latex]\\left(-\\infty ,0\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"\u201ctextbox\u201d\">\n<h2 class=\"mceTemp\">Graphic Piecewise-Defined Functions<\/h2>\n<p>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, <strong>absolute value<\/strong> can be defined as the <strong>magnitude<\/strong>, or <strong>modulus<\/strong>, 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=\"equation unnumbered\" style=\"text-align: center\">[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=\"equation unnumbered\" style=\"text-align: center\">[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 <strong>piecewise function<\/strong> 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 &#8220;boundaries.&#8221; 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, S, would be\u00a00.1S if [latex]{S}\\le\\[\/latex] $10,000\u00a0and 1000 + 0.2 (S &#8211; $10,000),\u00a0if S&gt; $10,000.<\/p>\n<div id=\"fs-id1165137531241\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Piecewise Function<\/h3>\n<p id=\"fs-id1165135504970\">A piecewise function 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<p style=\"text-align: center\">[latex]f\\left(x\\right)=\\begin{cases}\\text{formula 1 if x is in domain 1}\\\\ \\text{formula 2 if x is in domain 2}\\\\ \\text{formula 3 if x is in domain 3}\\end{cases}[\/latex]<\/p>\n<p>In piecewise notation, the absolute value function is<\/p>\n<p style=\"text-align: center\">[latex]|x|=\\begin{cases}\\begin{align}&x&&\\text{ if }x\\ge 0\\\\ &-x&&\\text{ if }x<0\\end{align}\\end{cases}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137768426\" class=\"note precalculus howto textbox\">\n<h3>How To:\u00a0Given a piecewise function, write the formula and identify the domain for each interval.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165135443772\">\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 shaded\">\n<h3>Example 11: Writing a Piecewise Function<\/h3>\n<p>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 <strong>function<\/strong> relating the number of people, [latex]n[\/latex], to the cost, [latex]C[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525510\">Show Solution<\/span><\/p>\n<div id=\"q525510\" class=\"hidden-answer\" style=\"display: none\">\n<p>Two different formulas will be needed. For <em>n<\/em>-values under 10, C=5n. For values of n that are 10 or greater, C=50.<\/p>\n<p style=\"text-align: center\">[latex]C(n)=\\begin{cases}\\begin{align}{5n}&\\hspace{5mm}\\text{ if }{0}<{n}<{10}\\\\ 50&\\hspace{5mm}\\text{ if }{n}\\ge 10\\end{align}\\end{cases}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The function is represented in Figure 21. 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 style=\"width: 370px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0212.jpg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 21<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex 2:  Graph a Piecewise Defined Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B1jfpiI-QQ8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"Example_01_02_12\" class=\"example\">\n<div id=\"fs-id1165135436662\" class=\"exercise\">\n<div id=\"fs-id1165135436664\" class=\"problem textbox shaded\">\n<h3>Example 12: 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=\"equation unnumbered\" style=\"text-align: center\">[latex]C\\left(g\\right)=\\begin{cases}\\begin{align}&{25} &&\\hspace{-5mm}\\text{ if }{ 0 }<{ g }<{ 2 }\\\\ &{ 25+10 }\\left(g - 2\\right) &&\\hspace{-5mm}\\text{ if }{ g}\\ge{ 2 }\\end{align}\\end{cases}[\/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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q67822\">Show Solution<\/span><\/p>\n<div id=\"q67822\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134373545\">To find the cost of using 1.5 gigabytes of data, C(1.5), 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<p style=\"text-align: center\">[latex]C(1.5) = \\$25[\/latex]<\/p>\n<p id=\"fs-id1165135440213\">To find the cost of using 4 gigabytes of data, C(4), we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\n<div style=\"text-align: center\">[latex]C(4)=25 + 10( 4-2) =\\$45[\/latex]<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>The function is represented in Figure 22. 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 style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010548\/CNX_Precalc_Figure_01_02_0222.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 22<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600493\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135532516\">How To:\u00a0Given a piecewise function, sketch a graph.<\/h3>\n<ol id=\"fs-id1165137588539\">\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 id=\"Example_01_02_13\" class=\"example\">\n<div id=\"fs-id1165137781618\" class=\"exercise\">\n<div id=\"fs-id1165135412870\" class=\"problem textbox shaded\">\n<h3>Example 13: Graphing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137838785\">Sketch a graph of the function.<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}&{ x }^{2} &&\\hspace{-5mm}\\text{ if }{ x }\\le{ 1 }\\\\ &{ 3 } &&\\hspace{-5mm}\\text{ if } { 1 }&lt{ x }\\le 2\\\\ &{ x } &&\\hspace{-5mm}\\text{ if }{ x }&gt{ 2 }\\end{align}\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617292\">Show Solution<\/span><\/p>\n<div id=\"q617292\" class=\"hidden-answer\" style=\"display: none\">\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\">Below are\u00a0the three components of the piecewise function graphed on separate coordinate systems.<\/p>\n<figure id=\"Figure_01_02_023\"><figcaption>(a) [latex]f\\left(x\\right)={x}^{2}\\text{ if }x\\le 1[\/latex]; (b) [latex]f\\left(x\\right)=3\\text{ if 1< }x\\le 2[\/latex]; (c) [latex]f\\left(x\\right)=x\\text{ if }x>2[\/latex]<\/figcaption><div style=\"width: 984px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010549\/CNX_Precalc_Figure_01_02_023abc2.jpg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 23<\/b><\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165137676209\">Now that we have sketched each piece individually, we combine them in the same coordinate plane.<span id=\"fs-id1165137646696\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010549\/CNX_Precalc_Figure_01_02_0262.jpg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 24<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/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]\u00a0and [latex]\\left(2,3\\right)[\/latex] are.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph the following piecewise function.<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=\\begin{cases}{ x}^{3} \\text{ if }{ x }&lt{-1 }\\\\ { -2 } \\text{ if } { -1 }&lt{ x }&lt{ 4 }\\\\ \\sqrt{x} \\text{ if }{ x }&gt{ 4 }\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q836663\">Show Solution<\/span><\/p>\n<div id=\"q836663\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010549\/CNX_Precalc_Figure_01_02_0272.jpg\" alt=\"Graph of f(x).\" width=\"487\" height=\"408\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm32883\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32883&theme=oea&iframe_resize_id=ohm32883\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165137810682\" class=\"note precalculus qa textbox\">\n<h3>Q&amp;A<\/h3>\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<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Key Concepts<\/span><\/h2>\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.<\/li>\n<li>The domain of a function can also be determined by identifying the input values of a function written as an equation.<\/li>\n<li>Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.<\/li>\n<li>For many functions, the domain and range can be determined from a graph.<\/li>\n<li>An understanding of toolkit functions can be used to find the domain and range of related functions.<\/li>\n<li>A piecewise function is described by more than one formula.<\/li>\n<li>A piecewise function can be graphed using each algebraic formula on its assigned subdomain.<\/li>\n<\/ul>\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Glossary<\/span><\/h2>\n<dl id=\"fs-id1165135445751\" class=\"definition\">\n<dt><strong>interval notation<\/strong><\/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\" class=\"definition\">\n<dt><strong>piecewise function<\/strong><\/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\" class=\"definition\">\n<dt><strong>set-builder notation<\/strong><\/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<dd><\/dd>\n<\/dl>\n<\/div>\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Section 1.6 Homework Exercises<\/span><\/h2>\n<p>1. Why does the domain differ for different functions?<\/p>\n<p>2. How do we determine the domain of a function defined by an equation?<\/p>\n<p>3. 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<p>4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?<\/p>\n<p>5. How do you graph a piecewise function?<\/p>\n<p id=\"fs-id1165137408926\">For the following exercises, find the domain of each function using interval notation.<\/p>\n<p>6. [latex]f\\left(x\\right)=-2x\\left(x - 1\\right)\\left(x - 2\\right)[\/latex]<\/p>\n<p>7. [latex]f\\left(x\\right)=5 - 2{x}^{2}[\/latex]<\/p>\n<p>8. [latex]f\\left(x\\right)=3\\sqrt{x - 2}[\/latex]<\/p>\n<p>9. [latex]f\\left(x\\right)=3-\\sqrt{6 - 2x}[\/latex]<\/p>\n<p>10. [latex]f\\left(x\\right)=\\sqrt{4 - 3x}[\/latex]<\/p>\n<p>11. [latex]f\\left(x\\right)=\\sqrt{{x}^{2}+4}[\/latex]<\/p>\n<p>12. [latex]f\\left(x\\right)=\\sqrt[3]{1 - 2x}[\/latex]<\/p>\n<p>13. [latex]f\\left(x\\right)=\\sqrt[3]{x - 1}[\/latex]<\/p>\n<p>14. [latex]f\\left(x\\right)=\\frac{9}{x - 6}[\/latex]<\/p>\n<p>15. [latex]f\\left(x\\right)=\\frac{3x+1}{4x+2}[\/latex]<\/p>\n<p>16. [latex]f\\left(x\\right)=\\frac{\\sqrt{x+4}}{x - 4}[\/latex]<\/p>\n<p>17. [latex]f\\left(x\\right)=\\frac{x - 3}{{x}^{2}+9x - 22}[\/latex]<\/p>\n<p>18. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-x - 6}[\/latex]<\/p>\n<p>19. [latex]f\\left(x\\right)=\\frac{2{x}^{3}-250}{{x}^{2}-2x - 15}[\/latex]<\/p>\n<p>20. [latex]\\frac{5}{\\sqrt{x - 3}}[\/latex]<\/p>\n<p>21. [latex]\\frac{2x+1}{\\sqrt{5-x}}[\/latex]<\/p>\n<p>22. [latex]f\\left(x\\right)=\\frac{\\sqrt{x - 4}}{\\sqrt{x - 6}}[\/latex]<\/p>\n<p>23. [latex]f\\left(x\\right)=\\frac{\\sqrt{x - 6}}{\\sqrt{x - 4}}[\/latex]<\/p>\n<p>24. [latex]f\\left(x\\right)=\\frac{x}{x}[\/latex]<\/p>\n<p>25. [latex]f\\left(x\\right)=\\frac{{x}^{2}-9x}{{x}^{2}-81}[\/latex]<\/p>\n<p>26. Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{2{x}^{3}-50x}[\/latex] by:<\/p>\n<div style=\"margin: 0 0 0 40px;border: none;padding: 0px\">a. using algebra.<br \/>\nb. graphing the function in the radicand and determining intervals on the <em>x<\/em>-axis for which the radicand is nonnegative.<\/div>\n<div><\/div>\n<div><\/div>\n<p>For the following exercises, write the domain and range of each function using interval notation.<\/p>\n<p>27.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005038\/CNX_Precalc_Figure_01_02_202.jpg\" alt=\"Graph of a function from (2, 8].\" width=\"487\" height=\"222\" \/><\/p>\n<p>Domain: ________ Range: ________<\/p>\n<p>28.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005038\/CNX_Precalc_Figure_01_02_203.jpg\" alt=\"Graph of a function from [4, 8).\" width=\"487\" height=\"222\" \/><\/p>\n<p>29.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005038\/CNX_Precalc_Figure_01_02_204.jpg\" alt=\"Graph of a function from [-4, 4].\" width=\"487\" height=\"220\" \/><\/p>\n<p>30.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1165135245908\" class=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_205.jpg\" alt=\"Graph of a function from [2, 6].\" width=\"487\" height=\"282\" \/><\/p>\n<p>31.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_206.jpg\" alt=\"Graph of a function from [-5, 3).\" width=\"487\" height=\"189\" \/><\/p>\n<p>32.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_207.jpg\" alt=\"Graph of a function from [-3, 2).\" width=\"487\" height=\"377\" \/><\/p>\n<p>33.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005039\/CNX_Precalc_Figure_01_02_208.jpg\" alt=\"Graph of a function from (-infinity, 2].\" width=\"487\" height=\"220\" \/><\/p>\n<p>34.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_209.jpg\" alt=\"Graph of a function from [-4, infinity).\" width=\"487\" height=\"316\" \/><\/p>\n<p>35.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_210.jpg\" alt=\"Graph of a function from [-6, -1\/6]U[1\/6, 6]\/.\" width=\"975\" height=\"442\" \/><\/p>\n<p>36.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_211.jpg\" alt=\"Graph of a function from (-2.5, infinity).\" width=\"487\" height=\"535\" \/><\/p>\n<p>37.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005040\/CNX_Precalc_Figure_01_02_212.jpg\" alt=\"Graph of a function from [-3, infinity).\" width=\"975\" height=\"379\" \/><\/p>\n<p>For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\n<p>38. [latex]f(x)=\\begin{cases}{x}+{1}&\\text{ if }&{ x }<{ -2 } \\\\{-2x - 3}&\\text{ if }&{ x }\\ge { -2 }\\\\ \\end{cases}[\/latex]\n\n39. [latex]f\\left(x\\right)=\\begin{cases}{2x - 1}&\\text{ if }&{ x }<{ 1 }\\\\ {1+x }&\\text{ if }&{ x }\\ge{ 1 } \\end{cases}[\/latex]\n\n40. [latex]f\\left(x\\right)=\\begin{cases}{x+1}&\\text{ if }&{ x }<{ 0 }\\\\ {x - 1 }&\\text{ if }&{ x }>{ 0 }\\end{cases}[\/latex]<\/p>\n<p>41. [latex]f\\left(x\\right)=\\begin{cases}{3} &\\text{ if }&{ x } <{ 0 }\\\\ \\sqrt{x}&\\text{ if }&{ x }\\ge { 0 }\\end{cases}[\/latex]\n\n42. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}&\\text{ if }&{ x } <{ 0 }\\\\ {1-x}&\\text{ if }&{ x } >{ 0 }\\end{cases}[\/latex]<\/p>\n<p>43. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}&\\text{ if }&{ x }<{ 0 }\\\\ {x+2 }&\\text{ if }&{ x }\\ge { 0 }\\end{cases}[\/latex]\n\n44. [latex]f\\left(x\\right)=\\begin{cases}x+1& \\text{if}& x<1\\\\ {x}^{3}& \\text{if}& x\\ge 1\\end{cases}[\/latex]\n\n45. [latex]f\\left(x\\right)=\\begin{cases}|x|&\\text{ if }&{ x }<{ 2 }\\\\ { 1 }&\\text{ if }&{ x }\\ge{ 2 }\\end{cases}[\/latex]\nFor 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].\n\n46. [latex]f\\left(x\\right)=\\begin{cases}{ x+1 }&\\text{ if }&{ x }<{ -2 }\\\\ { -2x - 3 }&\\text{ if }&{ x }\\ge{ -2 }\\end{cases}[\/latex]\n\n47. [latex]f\\left(x\\right)=\\begin{cases}{ 1 }&\\text{ if }&{ x }\\le{ -3 }\\\\{ 0 }&\\text{ if }&{ x }>{ -3 }\\end{cases}[\/latex]<\/p>\n<p>48. [latex]f\\left(x\\right)=\\begin{cases}{-2}{x}^{2}+{ 3 }&\\text{ if }&{ x }\\le { -1 }\\\\ { 5x } - { 7 } &\\text{ if }&{ x } > { -1 }\\end{cases}[\/latex]<br \/>\nFor 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<p>49. [latex]f\\left(x\\right)=\\begin{cases}{ 7x+3 }&\\text{ if }&{ x }<{ 0 }\\\\{ 7x+6 }&\\text{ if }&{ x }\\ge{ 0 }\\end{cases}[\/latex]\n\n50. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}{ -2 }&\\text{ if }&{ x }<{ 2 }\\\\{ 4+|x - 5|}&\\text{ if }&{ x }\\ge{ 2 }\\end{cases}[\/latex]\n\n51. [latex]f\\left(x\\right)=\\begin{cases}5x& \\text{if}& x<0\\\\ 3& \\text{if}& 0\\le x\\le 3\\\\ {x}^{2}& \\text{if}& x>3\\end{cases}[\/latex]<br \/>\nFor the following exercises, write the domain for the piecewise function in interval notation.<\/p>\n<p>52. [latex]f\\left(x\\right)=\\begin{cases}{x+1}&\\text{ if }&{ x }<{ -2 }\\\\{ -2x - 3}&\\text{ if }&{ x }\\ge{ -2 }\\end{cases}[\/latex]\n\n53. [latex]f\\left(x\\right)=\\begin{cases}{x}^{2}{ -2 }&\\text{ if}&{ x }<{ 1 }\\\\{-x}^{2}+{2}&\\text{ if }&{ x }>{ 1 }\\end{cases}[\/latex]<\/p>\n<p>54. [latex]f\\left(x\\right)=\\begin{cases}{ 2x - 3 }&\\text{ if }&{ x }<{ 0 }\\\\{ -3}{x}^{2}&\\text{ if }&{ x }\\ge{ 2 }\\end{cases}[\/latex]\n\n55. 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.\n\n56. 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.\n\n57. 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]\n\n58. Create a function in which the range is all nonnegative real numbers.\n\n59 .Create a function in which the domain is [latex]x>2[\/latex].<\/p>\n<p>60. 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<div style=\"margin: 0 0 0 40px;border: none;padding: 0px\">A. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.<br \/>\nB. What is the cost of making 25 items?<br \/>\nC. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, [latex]C\\left(x\\right)?[\/latex]<\/div>\n<p>61. 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]. What is the domain of the function? 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