{"id":38,"date":"2019-01-29T20:41:49","date_gmt":"2019-01-29T20:41:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=38"},"modified":"2025-03-31T20:46:44","modified_gmt":"2025-03-31T20:46:44","slug":"domain-and-range","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/domain-and-range\/","title":{"raw":"1.2 Domain and Range","rendered":"1.2 Domain and Range"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Use interval notation to express inequalities.<\/li>\r\n \t<li>Find the domain of a function defined by an equation, a graph or in context.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137404978\">If you\u2019re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time\u2014<em>I am Legend<\/em>, <em>Hannibal<\/em>, <em>The Ring<\/em>, <em>The Grudge<\/em>, and <em>The Conjuring<\/em>. <a class=\"autogenerated-content\" href=\"#Figure_01_02_001\">Figure 1<\/a> shows the amount, in dollars, each of those movies grossed when they were released as well as the percent of ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <span class=\"no-emphasis\">domain<\/span> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3077\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3077\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151818\/f29850a56c5a5ec5355fe59ed2e7ce1a3bb85627.jpeg\" alt=\"Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year\" width=\"975\" height=\"402\" \/> <strong>Figure 1\u00a0<\/strong>[\/caption]\r\n\r\n<div id=\"fs-id1165135193832\" class=\"bc-section section\">\r\n\r\nBased on data compiled by www.the-numbers.com.[footnote]The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d <a href=\"http:\/\/www.the-numbers.com\/market\/genre\/Horror\">http:\/\/www.the-numbers.com\/market\/genre\/Horror<\/a>. Accessed 3\/24\/2014[\/footnote]\r\n<h3>Finding the Domain of a Function Defined by an Equation<\/h3>\r\n<p id=\"fs-id1165135445896\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/functions-and-function-notation\/\">Section 1.1, Functions and Function Notation<\/a>, we were introduced to the concepts of <span class=\"no-emphasis\">domain and range<\/span>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\r\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \u201cholding area\u201d that contains \u201craw materials\u201d for a \u201cfunction machine\u201d and the range as another \u201cholding area\u201d for the machine\u2019s products. See <a class=\"autogenerated-content\" href=\"#Figure_01_02_002\">Figure 2<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3079\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3079\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151941\/9aef948122bea6a402035bb72d6cd277b4c97cf3.jpeg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/> Figure 2[\/caption]\r\n\r\n<div id=\"Figure_01_02_002\" class=\"small\"><\/div>\r\n<p id=\"fs-id1165137761714\">For functions of real numbers, we can write the <span class=\"no-emphasis\">domain and range<\/span> in interval notation, which uses values within brackets to describe a set of real numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write [latex]\\left(0,\\text{ }100\\right].[\/latex] We will discuss interval notation in greater detail later.<\/p>\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 an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root,\u00a0 exclude values that would make the radicand negative.<\/p>\r\n<p id=\"fs-id1165137552233\">Before we begin, let us discuss the conventions of interval notation:<\/p>\r\n\r\n<ul id=\"fs-id1165135673417\">\r\n \t<li>The smallest value from the interval is written first, followed by a comma.<\/li>\r\n \t<li>The largest value in the interval is written second.<\/li>\r\n \t<li>Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.<\/li>\r\n \t<li>Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.<\/li>\r\n \t<li>Use a symbol known as the union, [latex]\\cup ,[\/latex] to combine intervals if there are more than one.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137807384\">See <a class=\"autogenerated-content\" href=\"#Figure_01_02_029\">Figure 3<\/a> for a summary of interval notation.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3080\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3080\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152036\/b8e583b5d39ad28fd3c67fc29332cc49b19c1cd7.jpeg\" alt=\"Summary of interval notation.\" width=\"975\" height=\"905\" \/> Figure 3[\/caption]\r\n\r\n<div id=\"Example_01_02_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137661548\">\r\n<div id=\"fs-id1165137772018\">\r\n<h3>Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\r\n<p id=\"fs-id1165137920768\">Find the <span class=\"no-emphasis\">domain<\/span> of the following function: [latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135329797\">[reveal-answer q=\"fs-id1165135329797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135329797\"]\r\n<p id=\"fs-id1165135508343\">First identify the input values. The input value is the first coordinate in an <span class=\"no-emphasis\">ordered pair<\/span>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\r\n\r\n<div id=\"fs-id1165137451888\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"unnumbered\">We cannot use interval notation here, as we are not including all real numbers between 2 and 6.[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137569901\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"fs-id1165135333722\">\r\n<div id=\"fs-id1165137852040\">\r\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\r\n<p id=\"fs-id1165137466017\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137501477\">[reveal-answer q=\"fs-id1165137501477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137501477\"]\r\n<p id=\"fs-id1165137704712\">[latex]\\left\\{-5,\\text{ }0,\\text{ }5,\\text{ }10,\\text{ }15\\right\\}[\/latex]\u00a0 Again, we cannot use interval notation here.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134225655\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165134355557\"><strong>Given a function written in equation form, find the domain.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134187286\" type=\"1\">\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 id=\"Example_01_02_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137767649\">\r\n<div id=\"fs-id1165137761307\">\r\n<h3>Example 2:\u00a0 Finding the Domain of a Function<\/h3>\r\n<p id=\"fs-id1165137645656\">Find the domain of the function [latex]f\\left(x\\right)={x}^{2}-1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135684349\">[reveal-answer q=\"fs-id1165135684349\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135684349\"]\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\nNotice that we use parenthesis if we are working with an infinite set of values in both directions from zero.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135639906\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"fs-id1165137733850\">\r\n<div id=\"fs-id1165137871971\">\r\n<p id=\"fs-id1165137871972\">Find the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137809848\">[reveal-answer q=\"fs-id1165137809848\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137809848\"]\r\n<p id=\"fs-id1165137809849\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex].\u00a0 We know that any real number can be cubed, and have any other real numbers added or subtracted to it.\u00a0 This set of operations will produce a real number.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137417188\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137473617\"><strong>Given a function written in an equation form that includes a fraction, find the domain.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137463251\" type=\"1\">\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].<\/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 id=\"Example_01_02_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137722406\">\r\n<div id=\"fs-id1165135484119\">\r\n<h3>Example 3:\u00a0 Finding the Domain of a Function Involving a Denominator<\/h3>\r\n<p id=\"fs-id1165137647592\">Find the <span class=\"no-emphasis\">domain<\/span> of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135641743\">[reveal-answer q=\"fs-id1165135641743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135641743\"]\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=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}2-x&amp;=0\\\\-x&amp;=-2\\\\x&amp;=2\\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135192763\">Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x&lt;2[\/latex] or [latex]x&gt;2.[\/latex] 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>\r\n\r\n\r\n[caption id=\"attachment_3081\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3081\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152219\/f5b60423d3e7587eb268720f0b2b70524a8b80ed.jpeg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/> Figure 4[\/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\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133349280\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"fs-id1165137437630\">\r\n<div id=\"fs-id1165137771815\">\r\n<p id=\"fs-id1165137442339\">Find the domain of the function: [latex]f\\left(x\\right)=\\frac{1+4x}{2x-1}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137436024\">[reveal-answer q=\"fs-id1165137436024\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137436024\"]\r\n<p id=\"fs-id1165135186314\">[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex].\u00a0 We cannot let the denominator be equal to 0.\u00a0 This occurs when\u00a0[latex] x = \\frac{1}{2}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135527005\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137733733\"><strong>Given a function written in equation form including an even root, find the domain.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137820030\" type=\"1\">\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 id=\"Example_01_02_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135160109\">\r\n<div id=\"fs-id1165137735699\">\r\n<h3>Example 4:\u00a0 Finding the Domain of a Function with an Even Root<\/h3>\r\n<p id=\"fs-id1165137466144\">Find the <span class=\"no-emphasis\">domain<\/span> of the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{7-x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137451129\">[reveal-answer q=\"fs-id1165137451129\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137451129\"]\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=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}7-x&amp;\\ge 0\\\\-x&amp;\\ge -7\\\\x&amp;\\le 7\\hfill \\end{align*}[\/latex]<\/div>\r\n<div><\/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\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137737842\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"fs-id1165137933139\">\r\n<div id=\"fs-id1165137933140\">\r\n<p id=\"fs-id1165137452448\">Find the domain of the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{5+2x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137832331\">[reveal-answer q=\"fs-id1165137832331\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832331\"]\r\n<p id=\"fs-id1165137832332\">[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex].\u00a0 We know the radicand must be greater than or equal to 0.\u00a0 We find this occurs for values where the input is greater than or equal to [latex]-\\frac{5}{2}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134328219\" class=\"precalculus qa key-takeaways\">\r\n<h3>Q&amp;A<\/h3>\r\n<p id=\"fs-id1165137659456\"><strong>Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\r\n<p id=\"fs-id1165137937737\"><em>Yes. For example, the function [latex]f\\left(x\\right)=-\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{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\n<\/div>\r\n<div id=\"fs-id1165137677916\" class=\"bc-section section\">\r\n<h3>Using Notations to Specify Domain and Range<\/h3>\r\n<p id=\"fs-id1165137410091\">In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex] describes the behavior of [latex]x[\/latex] in set-builder notation. The braces [latex]\\left\\{\\right\\}[\/latex] are read as \u201cthe set of,\u201d and the vertical bar | is read as \u201csuch that,\u201d so we would read [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex] as \u201cthe set of <em>x<\/em>-values such that 10 is less than or equal to [latex]x,[\/latex] and [latex]x[\/latex] is less than 30.\u201d<\/p>\r\n<p id=\"fs-id1165135207589\"><a class=\"autogenerated-content\" href=\"#Figure_01_02_003\">Figure 5<\/a> compares inequality notation, set-builder notation, and interval notation.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3082\" align=\"aligncenter\" width=\"975\"]<img class=\"wp-image-3082 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152403\/832bf155f3aaad1b9dae947ac7c2d9927bc02938.jpeg\" alt=\"Summary of notations for inequalities, set-builder, and intervals.\" width=\"975\" height=\"692\" \/> Figure 5[\/caption]\r\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \u201cor.\u201d As we saw in earlier examples, we use the union symbol, [latex]\\cup ,[\/latex] to combine two unconnected intervals. For example, the union of the sets [latex]\\left\\{2,3,5\\right\\}[\/latex] and [latex]\\left\\{4,6\\right\\}[\/latex] is 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=\"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\n<div id=\"fs-id1165137641795\">\r\n<h3>Set-Builder Notation and Interval Notation<\/h3>\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137663670\"><strong>Set-builder notation <\/strong>is a method of specifying a set of elements that satisfy a certain condition. It takes the form [latex]\\left\\{x|\\text{ }\\text{statement about }x\\right\\}[\/latex] which is read as, \u201cthe set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.\u201d For example,<\/p>\r\n\r\n<div id=\"fs-id1165137543047\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{x|4\\lt x\\le 12\\right\\}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135190272\"><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example, [latex]\\left(4,12\\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137805770\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137423878\"><strong>Given a line graph, describe the set of values using interval notation.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134032280\" type=\"1\">\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 \\text{, }[\/latex]to combine all intervals into one set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_02_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134342702\">\r\n<div id=\"fs-id1165137803670\">\r\n<h3>Example 5:\u00a0 Describing Sets on the Real-Number Line<\/h3>\r\n<p id=\"fs-id1165137592069\">Describe the intervals of values shown in <a class=\"autogenerated-content\" href=\"#Figure_01_02_004\">Figure 6<\/a> using inequality notation, set-builder notation, and interval notation.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3083\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3083\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152546\/13c74d3bc393a72d003d8fa46c769c591786bc87-1.jpeg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/> Figure 6[\/caption]\r\n\r\n<\/div>\r\n<div class=\"small\"><\/div>\r\n<div id=\"fs-id1165135412904\">[reveal-answer q=\"fs-id1165135412904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135412904\"]\r\n<p id=\"fs-id1165135412905\">To describe the values, [latex]x,[\/latex] included in the intervals shown, we would say, \u201c[latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\u201d<\/p>\r\n\r\n<table id=\"fs-id1165137447518\" class=\"unnumbered\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>Inequality<\/strong><\/td>\r\n<td class=\"border\">[latex]1\\le x\\le 3\\text{ }\\text{or}\\text{ }x&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>Set-builder notation<\/strong><\/td>\r\n<td class=\"border\">[latex]\\left\\{x|1\\le x\\le 3\\text{ }\\text{or}\\text{ }x&gt;5\\right\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>Interval notation<\/strong><\/td>\r\n<td class=\"border\">[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 id=\"fs-id1165137779165\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"ti_01_02_03\">\r\n<div id=\"fs-id1165135175087\">\r\n<p id=\"fs-id1165135341412\">Given <a class=\"autogenerated-content\" href=\"#Figure_01_02_005\">Figure 7<\/a>, specify the graphed set in<\/p>\r\n\r\n<ol id=\"fs-id1165137595582\" type=\"a\">\r\n \t<li>words<\/li>\r\n \t<li>set-builder notation<\/li>\r\n \t<li>interval notation\r\n\r\n[caption id=\"attachment_3084\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3084\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152753\/eba2f61d5fb32eca2dd57b7de1a1e57511a15f6a.jpeg\" alt=\"\" width=\"487\" height=\"50\" \/> Figure 7[\/caption]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"small\"><\/div>\r\n<div id=\"ti_01_02_03\">\r\n<div id=\"fs-id1165135209390\">[reveal-answer q=\"fs-id1165135209390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135209390\"]\r\n<ol id=\"fs-id1165135528963\" type=\"a\">\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{ }\\text{or}\\text{ }-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\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137653855\" class=\"bc-section section\">\r\n<h3>Finding Domain and Range from Graphs<\/h3>\r\n<p id=\"fs-id1165135161404\">Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em>x<\/em>-axis. The range is the set of possible output values, which are shown on the <em>y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. This is indicated by the arrows in\u00a0<a class=\"autogenerated-content\" href=\"#Figure_01_02_006\">Figure 8<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3085\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3085 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152837\/aec6dfffbf1ae7c213cf7c467e97eade263a24c9.jpeg\" alt=\"\" width=\"487\" height=\"666\" \/> Figure 8[\/caption]\r\n\r\n<div id=\"Figure_01_02_006\" class=\"small\"><\/div>\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 lower to higher 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=\"textbox examples\">\r\n<div id=\"fs-id1165137561401\">\r\n<div id=\"fs-id1165137599824\">\r\n<h3>Example 6:\u00a0 Finding Domain and Range from a Graph<\/h3>\r\n[caption id=\"attachment_3086\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3086 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152944\/940d0e7a8d6cc139b44a525718f304d15c48916b.jpeg\" alt=\"\" width=\"487\" height=\"364\" \/> Figure 9[\/caption]\r\n<p id=\"fs-id1165135187604\">Find the domain and range of the function [latex]f[\/latex] whose graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_02_007\">Figure 9<\/a>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"small\"><\/div>\r\n<div id=\"fs-id1165137561401\">\r\n<div id=\"fs-id1165137575085\">[reveal-answer q=\"fs-id1165137575085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137575085\"]\r\n<p id=\"fs-id1165137768165\">We can observe that the horizontal extent of the graph is \u20133 but not inclusive to 1 included, so the domain of [latex]f[\/latex] is [latex]\\left(-3,1\\right].[\/latex]<\/p>\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] See <a class=\"autogenerated-content\" href=\"#Figure_01_02_008\">Figure 10<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_02_008\" class=\"small\">\r\n\r\n[caption id=\"attachment_3087\" align=\"aligncenter\" width=\"488\"]<img class=\"size-full wp-image-3087\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153050\/258d8eebdb402305613f6b1ee6cfb9c3b0e16c0f.jpeg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"488\" height=\"364\" \/> Figure 10[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_02_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134182686\">\r\n<div id=\"fs-id1165137461643\">\r\n<h3>Example 7:\u00a0 Finding Domain and Range from a Graph of Oil Production<\/h3>\r\n<p id=\"fs-id1165137443324\">Find the domain and range of the function [latex]f[\/latex] whose graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_02_009\">Figure\u00a0 11<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3088\" align=\"aligncenter\" width=\"489\"]<img class=\"size-full wp-image-3088\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153150\/e7beda4d06fbbe861ed77bf9eb06f3823c8932a7.jpeg\" 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\" \/> Figure 11[\/caption]\r\n\r\n<div id=\"Figure_01_02_009\" class=\"small\"><\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div>(credit: modification of work by the U.S. Energy Information Administration)[footnote]<a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A\">http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A<\/a>.[\/footnote]<\/div>\r\n<div id=\"fs-id1165137444311\">[reveal-answer q=\"fs-id1165137444311\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137444311\"]\r\n<p id=\"fs-id1165137476085\">The input quantity along the horizontal axis is \u201cyears,\u201d which we represent with the variable [latex]t[\/latex] for time. The output quantity is \u201cthousands of barrels of oil per day,\u201d which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010.[\/latex]<\/p>\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 id=\"fs-id1165135545972\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"ti_01_02_04\">\r\n<div id=\"fs-id1165137644581\">\r\n<p id=\"fs-id1165137644582\">Given <a class=\"autogenerated-content\" href=\"#Figure_01_02_010\">Figure 12<\/a>, identify the domain and range using interval notation.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3091\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3091 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153447\/d86a2e251962e6baf7039f5a05db37e4bb123d1a.jpeg\" 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\" \/> Figure 12[\/caption]\r\n\r\n<div id=\"Figure_01_02_010\" class=\"small\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137705252\">[reveal-answer q=\"fs-id1165137705252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137705252\"]\r\n<p id=\"fs-id1165134079741\">domain = [1950,2002] range = [47,000,000,89,000,000]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137434590\" class=\"precalculus qa key-takeaways\">\r\n<h3>Q&amp;A<\/h3>\r\n<p id=\"fs-id1165137812796\"><strong>Can a function\u2019s domain and range be the same?<\/strong><\/p>\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<\/div>\r\n<div id=\"fs-id1165134384565\" class=\"bc-section section\">\r\n<h3>Finding Domains and Ranges of the Toolkit Functions<\/h3>\r\n<p id=\"fs-id1165137419914\">We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\r\n\r\n[caption id=\"attachment_3093\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3093\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153618\/7288c31bf0e35bb4bfc8a17e8a2320aa037d56a9.jpeg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/> For the constant function [latex]f\\left(x\\right)=c,[\/latex] the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c,[\/latex] so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right],[\/latex] the interval that both begins and ends with [latex]c.[\/latex][\/caption]\r\n<div id=\"Figure_01_02_011\" class=\"small\"><\/div>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_3094\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3094\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14154010\/71563b67e61e6ec6a9736686b246db9533b52ddf.jpeg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/> For the identity function [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.[\/caption]\u00a0[caption id=\"attachment_3095\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3095\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14154302\/f38fb33b0045eeb53fa467758aad11565d334626.jpeg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/> For the absolute value function [latex]f\\left(x\\right)=|x|,[\/latex] there is no restriction on [latex]x.[\/latex] However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.[\/caption]\r\n<div id=\"Figure_01_02_013\" class=\"small\"><\/div>\r\n<div class=\"wp-caption-text\"><\/div>\r\n[caption id=\"attachment_3096\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3096 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14154936\/babe50d816f29913cd05a7b8e4c9cc7564fea57d.jpeg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/> For the quadratic function [latex]f\\left(x\\right)={x}^{2},[\/latex] the domain is all real numbers since any real number can be squared and result in a real number.\u00a0 We can also see graphically that the horizontal extent of the graph is the whole real number line. We can also see vertically, the graph does not include any negative values for the range, so the range is only nonnegative real numbers.[\/caption]\u00a0[caption id=\"attachment_3097\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3097\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155123\/2cf19d06d7beae28a6d80ed0898a7f43e6c32d86.jpeg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/> For the cubic function [latex]f\\left(x\\right)={x}^{3},[\/latex] the domain is all real numbers because any real number raised to the third power will result in a real number.\u00a0 Graphically we can see that the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.[\/caption]\r\n<div id=\"Figure_01_02_014\" class=\"small\"><\/div>\r\n<div id=\"Figure_01_02_015\" class=\"small\"><\/div>\r\n[caption id=\"attachment_3098\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3098 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155520\/44f1be11fdaa6b420f329bebb3a745e40531e363.jpeg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/> 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. The graph above indicates all other real numbers can be included in the range. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\},[\/latex] the set of all real numbers that are not zero.[\/caption]\r\n<div id=\"Figure_01_02_016\" class=\"small\"><\/div>\r\n<div class=\"wp-caption-text\"><\/div>\r\n[caption id=\"attachment_3099\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3099 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155648\/7cac5fab3a66d73e68580db939aa2513f077076e.jpeg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/> For the reciprocal squared function [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}},[\/latex] we cannot divide by [latex]0,[\/latex] so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.[\/caption]\r\n<div id=\"Figure_01_02_017\" class=\"small\"><\/div>\r\n[caption id=\"attachment_3100\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3100\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155812\/edda7a5a94c145e7a95e5657060c0d0b08c5ccdf.jpeg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/> For the square root function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{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[\\leftroot{1}\\uproot{2} ]{x}[\/latex] also gives us [latex]x.[\/latex][\/caption]\r\n<div id=\"Figure_01_02_018\" class=\"small\"><\/div>\r\n[caption id=\"attachment_3101\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3101\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160003\/5e97fc8781f0bb16f912cac0d584182c20d5ca5a.jpeg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/> For the cube root function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}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). We can see from the graph that we have not restrictions either horizontally or vertically.[\/caption]\r\n<div id=\"Figure_01_02_019\" class=\"small\"><\/div>\r\n<div class=\"wp-caption-text\"><\/div>\r\n<div id=\"fs-id1165137462732\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137611181\"><strong>Given the formula for a function, determine the domain and range.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137405229\" type=\"1\">\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=\"textbox examples\">\r\n<div id=\"fs-id1165137558723\">\r\n<div id=\"fs-id1165137464274\">\r\n<h3>Example 8:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165135458670\">[reveal-answer q=\"fs-id1165135458670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135458670\"]\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.\u00a0 The range can be found by graphing the function and viewing the vertical extent of the graph.\u00a0 The range of this type of function will be discussed in detail in chapter 4.<\/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=\"textbox examples\">\r\n<div id=\"fs-id1165137448155\">\r\n<div id=\"fs-id1165137661316\">\r\n<h3>Example 9:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165137871182\">[reveal-answer q=\"fs-id1165137871182\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137871182\"]\r\n<p id=\"fs-id1165137855321\">We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex]\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right).[\/latex] Because the function is never zero, we exclude 0 from the range. The range is [latex]\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_02_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137574740\">\r\n<div id=\"fs-id1165135641583\">\r\n<h3>Example 10:\u00a0 Finding the Domain and Range<\/h3>\r\n<p id=\"fs-id1165137661054\">Find the domain and range of [latex]f\\left(x\\right)=2\\sqrt[\\leftroot{1}\\uproot{2} ]{x+4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137584342\">[reveal-answer q=\"fs-id1165137584342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137584342\"]\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=\"unnumbered\" style=\"text-align: center;\">[latex]x+4\\ge 0[\/latex] when [latex]x\\ge -4[\/latex]<\/div>\r\n<div><\/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<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165137437183\"><a class=\"autogenerated-content\" href=\"#Figure_01_02_020\">Figure 13<\/a> represents the function [latex]f.[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_3103\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3103\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160248\/d001600799023600e805aeb76b812fb5501ea12d-1.jpeg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/> Figure 13[\/caption]\r\n\r\n<div id=\"Figure_01_02_020\" class=\"small\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137572635\">\r\n<div id=\"Figure_01_02_020\" class=\"small\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137430800\" class=\"precalculus tryit\">\r\n<h3>Try it #7<\/h3>\r\n<div id=\"ti_01_02_05\">\r\n<div id=\"fs-id1165137475544\">\r\n<p id=\"fs-id1165137475545\">Find the domain and range of [latex]f\\left(x\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{2-x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137833252\">[reveal-answer q=\"fs-id1165137833252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137833252\"]\r\n<p id=\"fs-id1165137725047\">domain: [latex]\\left(-\\infty ,2\\right];[\/latex] range: [latex]\\left(-\\infty ,0\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135440477\" class=\"bc-section section\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 11:\u00a0 Find the Domain<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{x-1}}.[\/latex]\r\n\r\n[reveal-answer q=\"682389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"682389\"]\r\n\r\nThis function involves both a fraction and an even root so we need to consider how each of these properties will effect the domain.\u00a0 First we know that the expression under the radicand, [latex]x-1,[\/latex] needs to be greater than or equal to zero.\u00a0 However, if the expression equals zero, we will get a divide by zero.\u00a0 Therefore, because the square root is in the denominator, we solve\r\n<p style=\"text-align: center;\">[latex]x-1&gt;0.[\/latex]<\/p>\r\nThis simplifies to [latex]x&gt;1,[\/latex] so the domain is [latex]\\left(1,\\text{ }\\infty\\right).[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Graphing Piecewise-Defined Functions (Optional)<\/h3>\r\n<p id=\"fs-id1165137409262\">Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function [latex]f\\left(x\\right)=|x|.[\/latex] With a domain of all real numbers and a range of values greater than or equal to 0, <span class=\"no-emphasis\">absolute value<\/span> can be defined as the <span class=\"no-emphasis\">magnitude<\/span>, or <span class=\"no-emphasis\">modulus<\/span>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\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=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=x\\text{ }\\text{if}\\text{ }x\\ge 0[\/latex][latex]\\\\[\/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=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-x\\text{ }\\text{if}\\text{ }x&lt;0[\/latex][latex]\\\\[\/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.\u00a0 A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\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 \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income [latex]S[\/latex] would be [latex]0.1S[\/latex] if\u00a0[latex]S\\le 10\\text{,}000[\/latex] and [latex]1000+0.2\\left(S-10\\text{,}000\\right)[\/latex] if [latex]S&gt;10\\text{,}000.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137531241\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165135504970\">A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:<\/p>\r\n\r\n<div id=\"fs-id1165137482244\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{l}\\text{formula 1 if }x\\text{ is in domain 1}\\\\ \\text{formula 2 if }x\\text{ is in domain 2}\\\\ \\text{formula 3 if }x\\text{ is in domain 3}\\end{array}[\/latex]<\/div>\r\n<\/div>\r\nIn piecewise notation, the absolute value function is\r\n<div id=\"fs-id1165135190749\" class=\"unnumbered\" style=\"text-align: center;\">[latex]|x|=\\bigg\\{\\begin{array}{l}x\\text{ if }x\\ge 0\\\\ -x\\text{ if }x&lt;0\\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137768426\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137823161\"><strong>Given a piecewise function, write the formula and identify the domain for each interval. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135443772\" type=\"1\">\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 id=\"Example_01_02_11\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137452506\">\r\n<div id=\"fs-id1165135321994\">\r\n<h3>Example 12:\u00a0 Writing a Piecewise Function<\/h3>\r\n<p id=\"fs-id1165137834905\">A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <span class=\"no-emphasis\">function<\/span> relating the number of people, [latex]n,[\/latex] to the cost, [latex]C.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137807421\">[reveal-answer q=\"fs-id1165137807421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137807421\"]\r\n<p id=\"fs-id1165135331729\">Two different formulas will be needed. For <em>n<\/em>-values under 10, [latex]C=5n.[\/latex] For values of [latex]n[\/latex] that are 10 or greater, [latex]C=50.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135208951\" class=\"unnumbered\" style=\"text-align: center;\">[latex]C\\left(n\\right)=\\bigg\\{\\begin{array}{ccc}5n&amp; \\text{if}&amp; 0\\lt n \\lt10\\\\ 50&amp; \\text{if}&amp; n\\ge 10\\end{array}[\/latex]<\/div>\r\n<div>\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165135196985\">The function is represented in <a class=\"autogenerated-content\" href=\"#Figure_01_02_021\">Figure 14<\/a>. The graph is a diagonal line from [latex]n=0[\/latex] to [latex]n=10[\/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10,[\/latex] but not all piecewise functions have this property.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3104\" align=\"aligncenter\" width=\"360\"]<img class=\"size-full wp-image-3104\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160457\/ddd5650076d9fc0801b73b14769d868d34a4b148-1.jpeg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/> Figure 14[\/caption]\r\n\r\n<div id=\"Figure_01_02_021\" class=\"small\"><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_02_12\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135436662\">\r\n<div id=\"fs-id1165135436664\">\r\n<h3>Example 13:\u00a0 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=\"unnumbered\" style=\"text-align: center;\">[latex]C\\left(g\\right)=\\bigg\\{\\begin{array}{ccc}25&amp; \\text{if}&amp; 0 \\lt g&lt;2\\\\ 25+10\\left(g-2\\right)&amp; \\text{if}&amp; g\\ge 2\\end{array}[\/latex]<\/div>\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\r\n<\/div>\r\n<div id=\"fs-id1165135177567\">[reveal-answer q=\"fs-id1165135177567\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135177567\"]\r\n<p id=\"fs-id1165134373545\">To find the cost of using 1.5 gigabytes of data, [latex]C\\left(1.5\\right),[\/latex] we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.<\/p>\r\n\r\n<div id=\"fs-id1165134300204\" style=\"text-align: center;\">[latex]C\\left(1.5\\right)=25[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135440213\">To find the cost of using 4 gigabytes of data, [latex]C\\left(4\\right),[\/latex] we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\r\n\r\n<div id=\"fs-id1165135383665\" style=\"text-align: center;\">[latex]C\\left(4\\right)=25+10\\left(4-2\\right)=45[\/latex]<\/div>\r\n<div><\/div>\r\n<div>\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165137601265\">The function is represented in <a class=\"autogenerated-content\" href=\"#Figure_01_02_022\">Figure 15<\/a>. We can see where the function changes from a constant to a linear\u00a0 function with slope 10 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>\r\n\r\n\r\n[caption id=\"attachment_3105\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3105\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160604\/75d850575e89cd19aeea1fb435b9ece4313fb816.jpeg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/> Figure 15[\/caption]\r\n\r\n<div id=\"Figure_01_02_022\" class=\"\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137600493\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135532516\"><strong>Given a piecewise function, sketch a graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137588539\" type=\"1\">\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=\"textbox examples\">\r\n<div id=\"fs-id1165137781618\">\r\n<div id=\"fs-id1165135412870\">\r\n<h3>Example 14:\u00a0 Graphing a Piecewise Function<\/h3>\r\n<p id=\"fs-id1165137838785\">Sketch a graph of the function.<\/p>\r\n\r\n<div id=\"fs-id1165137475346\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}{x}^{2}&amp; \\text{if}&amp; x\\le 1\\\\ 3&amp; \\text{if}&amp; 1\\lt x\\le 2\\\\ x&amp; \\text{if}&amp; x&gt;2\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135487148\">[reveal-answer q=\"fs-id1165135487148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135487148\"]\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<a class=\"autogenerated-content\" href=\"#Figure_01_02_023\">Figure 16<\/a> shows the three components of the piecewise function graphed on separate coordinate systems.\r\n\r\n[caption id=\"attachment_3106\" align=\"alignnone\" width=\"974\"]<img class=\"size-full wp-image-3106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160736\/8709146c78394619c275f4d39250c87b94329905.jpeg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/> Figure 16\u00a0(a) [latex]f\\left(x\\right)={x}^{2}\\text{ if }x\\le 1; [\/latex] (b) [latex]f\\left(x\\right)=3\\text{ if 1&lt; }x\\le 2; [\/latex] (c) [latex]f\\left(x\\right)=x\\text{ if }x&gt;2[\/latex][\/caption]\r\n<p id=\"fs-id1165137676209\">Now that we have sketched each piece individually, we combine them in the same coordinate plane. See <a class=\"autogenerated-content\" href=\"#Figure_01_02_026\">Figure 17<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_02_026\" class=\"small\">\r\n\r\n[caption id=\"attachment_3107\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3107\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160824\/e20a45f4db87b9669c40bc522d232a07b8147103-1.jpeg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/> Figure 17[\/caption]\r\n\r\n&nbsp;\r\n<h3>Analysis<\/h3>\r\nNote 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] and [latex]\\left(2,\\text{ }3\\right)[\/latex] are.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137762558\" class=\"precalculus tryit\">\r\n<h3>Try it #8<\/h3>\r\n<div id=\"ti_01_02_06\">\r\n<div id=\"fs-id1165137692562\">\r\n<p id=\"fs-id1165137692563\">Graph the following piecewise function.<\/p>\r\n\r\n<div id=\"fs-id1165137433350\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}{x}^{3}&amp; \\text{if}&amp; x \\lt -1\\\\ -2&amp; \\text{if}&amp; -1\\lt x\\lt 4\\\\ \\sqrt{x}&amp; \\text{if}&amp; x&gt;4\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137784656\">[reveal-answer q=\"fs-id1165137784656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137784656\"]<\/div>\r\n<div><span id=\"fs-id1165134302462\"><img class=\"alignnone size-full wp-image-3172\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/15144941\/55b54680118bbe7214755035f529a3ba2024d244.jpeg\" alt=\"Try it 8\" width=\"487\" height=\"408\" \/><\/span><\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137810682\" class=\"precalculus qa key-takeaways\">\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<div id=\"fs-id1165135190393\" class=\"precalculus media\">\r\n<p id=\"fs-id1165137627040\">Access these online resources for additional instruction and practice with domain and range.<\/p>\r\n\r\n<ul id=\"fs-id1165135189954\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/domainsqroot\">Domain and Range of Square Root Functions<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM&amp;feature=youtu.be%2F\r\n<ul id=\"fs-id1165135189954\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/determinedomain\">Determining Domain and Range<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=FtJRstFMdhA\r\n<ul id=\"fs-id1165135189954\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/drgraph\">Find Domain and Range Given the Graph<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=8jrkzZy04BQ&amp;feature=youtu.be%2F\r\n<ul id=\"fs-id1165135189954\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/drtable\">Find Domain and Range Given a Table<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=GPBq18fCEv4&amp;feature=youtu.be%2F\r\n<ul id=\"fs-id1165135189954\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/drcoordinate\">Find Domain and Range Given Points on a Coordinate Plane<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=xOsYVyjTM0Q&amp;feature=youtu.be%2F\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134077347\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\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 for functions of real numbers 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<\/ul>\r\nOptional:\r\n<ul id=\"fs-id1165137591772\">\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<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl>\r\n \t<dt>interval notation<\/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\">\r\n \t<dt>piecewise function<\/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\">\r\n \t<dt>set-builder notation<\/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{ }\\text{statement about }x\\right\\}[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Use interval notation to express inequalities.<\/li>\n<li>Find the domain of a function defined by an equation, a graph or in context.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137404978\">If you\u2019re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time\u2014<em>I am Legend<\/em>, <em>Hannibal<\/em>, <em>The Ring<\/em>, <em>The Grudge<\/em>, and <em>The Conjuring<\/em>. <a class=\"autogenerated-content\" href=\"#Figure_01_02_001\">Figure 1<\/a> shows the amount, in dollars, each of those movies grossed when they were released as well as the percent of ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <span class=\"no-emphasis\">domain<\/span> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.<\/p>\n<div id=\"attachment_3077\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3077\" class=\"size-full wp-image-3077\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151818\/f29850a56c5a5ec5355fe59ed2e7ce1a3bb85627.jpeg\" 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 id=\"caption-attachment-3077\" class=\"wp-caption-text\"><strong>Figure 1\u00a0<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1165135193832\" class=\"bc-section section\">\n<p>Based on data compiled by www.the-numbers.com.<a class=\"footnote\" title=\"The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014\" id=\"return-footnote-38-1\" href=\"#footnote-38-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<h3>Finding the Domain of a Function Defined by an Equation<\/h3>\n<p id=\"fs-id1165135445896\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/functions-and-function-notation\/\">Section 1.1, Functions and Function Notation<\/a>, we were introduced to the concepts of <span class=\"no-emphasis\">domain and range<\/span>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \u201cholding area\u201d that contains \u201craw materials\u201d for a \u201cfunction machine\u201d and the range as another \u201cholding area\u201d for the machine\u2019s products. See <a class=\"autogenerated-content\" href=\"#Figure_01_02_002\">Figure 2<\/a>.<\/p>\n<div id=\"attachment_3079\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3079\" class=\"size-full wp-image-3079\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14151941\/9aef948122bea6a402035bb72d6cd277b4c97cf3.jpeg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/><\/p>\n<p id=\"caption-attachment-3079\" class=\"wp-caption-text\">Figure 2<\/p>\n<\/div>\n<div id=\"Figure_01_02_002\" class=\"small\"><\/div>\n<p id=\"fs-id1165137761714\">For functions of real numbers, we can write the <span class=\"no-emphasis\">domain and range<\/span> in interval notation, which uses values within brackets to describe a set of real numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write [latex]\\left(0,\\text{ }100\\right].[\/latex] We will discuss interval notation in greater detail later.<\/p>\n<p id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root,\u00a0 exclude values that would make the radicand negative.<\/p>\n<p id=\"fs-id1165137552233\">Before we begin, let us discuss the conventions of interval notation:<\/p>\n<ul id=\"fs-id1165135673417\">\n<li>The smallest value from the interval is written first, followed by a comma.<\/li>\n<li>The largest value in the interval is written second.<\/li>\n<li>Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.<\/li>\n<li>Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.<\/li>\n<li>Use a symbol known as the union, [latex]\\cup ,[\/latex] to combine intervals if there are more than one.<\/li>\n<\/ul>\n<p id=\"fs-id1165137807384\">See <a class=\"autogenerated-content\" href=\"#Figure_01_02_029\">Figure 3<\/a> for a summary of interval notation.<\/p>\n<div id=\"attachment_3080\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3080\" class=\"size-full wp-image-3080\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152036\/b8e583b5d39ad28fd3c67fc29332cc49b19c1cd7.jpeg\" alt=\"Summary of interval notation.\" width=\"975\" height=\"905\" \/><\/p>\n<p id=\"caption-attachment-3080\" class=\"wp-caption-text\">Figure 3<\/p>\n<\/div>\n<div id=\"Example_01_02_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137661548\">\n<div id=\"fs-id1165137772018\">\n<h3>Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\n<p id=\"fs-id1165137920768\">Find the <span class=\"no-emphasis\">domain<\/span> of the following function: [latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135329797\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135329797\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135329797\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135508343\">First identify the input values. The input value is the first coordinate in an <span class=\"no-emphasis\">ordered pair<\/span>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\n<div id=\"fs-id1165137451888\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/div>\n<div><\/div>\n<div class=\"unnumbered\">We cannot use interval notation here, as we are not including all real numbers between 2 and 6.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137569901\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"fs-id1165135333722\">\n<div id=\"fs-id1165137852040\">\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\n<p id=\"fs-id1165137466017\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137501477\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137501477\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137501477\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137704712\">[latex]\\left\\{-5,\\text{ }0,\\text{ }5,\\text{ }10,\\text{ }15\\right\\}[\/latex]\u00a0 Again, we cannot use interval notation here.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134225655\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134355557\"><strong>Given a function written in equation form, find the domain.<\/strong><\/p>\n<ol id=\"fs-id1165134187286\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\n<li>Write the domain in interval form, if possible.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_02_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137767649\">\n<div id=\"fs-id1165137761307\">\n<h3>Example 2:\u00a0 Finding the Domain of a Function<\/h3>\n<p id=\"fs-id1165137645656\">Find the domain of the function [latex]f\\left(x\\right)={x}^{2}-1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135684349\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135684349\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135684349\" 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<p>Notice that we use parenthesis if we are working with an infinite set of values in both directions from zero.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135639906\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"fs-id1165137733850\">\n<div id=\"fs-id1165137871971\">\n<p id=\"fs-id1165137871972\">Find the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137809848\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137809848\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137809848\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137809849\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex].\u00a0 We know that any real number can be cubed, and have any other real numbers added or subtracted to it.\u00a0 This set of operations will produce a real number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137417188\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137473617\"><strong>Given a function written in an equation form that includes a fraction, find the domain.<\/strong><\/p>\n<ol id=\"fs-id1165137463251\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for [latex]x[\/latex].<\/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 id=\"Example_01_02_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137722406\">\n<div id=\"fs-id1165135484119\">\n<h3>Example 3:\u00a0 Finding the Domain of a Function Involving a Denominator<\/h3>\n<p id=\"fs-id1165137647592\">Find the <span class=\"no-emphasis\">domain<\/span> of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135641743\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135641743\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135641743\" 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=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}2-x&=0\\\\-x&=-2\\\\x&=2\\end{align*}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135192763\">Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[\/latex] or [latex]x>2.[\/latex] We can use a symbol known as the union, [latex]\\cup,[\/latex] to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right).[\/latex]<\/p>\n<div id=\"attachment_3081\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3081\" class=\"size-full wp-image-3081\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152219\/f5b60423d3e7587eb268720f0b2b70524a8b80ed.jpeg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/><\/p>\n<p id=\"caption-attachment-3081\" class=\"wp-caption-text\">Figure 4<\/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<\/div>\n<\/div>\n<div id=\"fs-id1165133349280\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"fs-id1165137437630\">\n<div id=\"fs-id1165137771815\">\n<p id=\"fs-id1165137442339\">Find the domain of the function: [latex]f\\left(x\\right)=\\frac{1+4x}{2x-1}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137436024\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137436024\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137436024\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135186314\">[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex].\u00a0 We cannot let the denominator be equal to 0.\u00a0 This occurs when\u00a0[latex]x = \\frac{1}{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135527005\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137733733\"><strong>Given a function written in equation form including an even root, find the domain.<\/strong><\/p>\n<ol id=\"fs-id1165137820030\" type=\"1\">\n<li>Identify the input values.<\/li>\n<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x.[\/latex]<\/li>\n<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_02_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135160109\">\n<div id=\"fs-id1165137735699\">\n<h3>Example 4:\u00a0 Finding the Domain of a Function with an Even Root<\/h3>\n<p id=\"fs-id1165137466144\">Find the <span class=\"no-emphasis\">domain<\/span> of the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{7-x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137451129\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137451129\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137451129\" 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=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}7-x&\\ge 0\\\\-x&\\ge -7\\\\x&\\le 7\\hfill \\end{align*}[\/latex]<\/div>\n<div><\/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<\/div>\n<\/div>\n<div id=\"fs-id1165137737842\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"fs-id1165137933139\">\n<div id=\"fs-id1165137933140\">\n<p id=\"fs-id1165137452448\">Find the domain of the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{5+2x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137832331\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137832331\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137832331\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137832332\">[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex].\u00a0 We know the radicand must be greater than or equal to 0.\u00a0 We find this occurs for values where the input is greater than or equal to [latex]-\\frac{5}{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134328219\" class=\"precalculus qa key-takeaways\">\n<h3>Q&amp;A<\/h3>\n<p id=\"fs-id1165137659456\"><strong>Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\n<p id=\"fs-id1165137937737\"><em>Yes. For example, the function [latex]f\\left(x\\right)=-\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137677916\" class=\"bc-section section\">\n<h3>Using Notations to Specify Domain and Range<\/h3>\n<p id=\"fs-id1165137410091\">In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example, [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex] describes the behavior of [latex]x[\/latex] in set-builder notation. The braces [latex]\\left\\{\\right\\}[\/latex] are read as \u201cthe set of,\u201d and the vertical bar | is read as \u201csuch that,\u201d so we would read [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex] as \u201cthe set of <em>x<\/em>-values such that 10 is less than or equal to [latex]x,[\/latex] and [latex]x[\/latex] is less than 30.\u201d<\/p>\n<p id=\"fs-id1165135207589\"><a class=\"autogenerated-content\" href=\"#Figure_01_02_003\">Figure 5<\/a> compares inequality notation, set-builder notation, and interval notation.<\/p>\n<div id=\"attachment_3082\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3082\" class=\"wp-image-3082 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152403\/832bf155f3aaad1b9dae947ac7c2d9927bc02938.jpeg\" alt=\"Summary of notations for inequalities, set-builder, and intervals.\" width=\"975\" height=\"692\" \/><\/p>\n<p id=\"caption-attachment-3082\" class=\"wp-caption-text\">Figure 5<\/p>\n<\/div>\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \u201cor.\u201d As we saw in earlier examples, we use the union symbol, [latex]\\cup ,[\/latex] to combine two unconnected intervals. For example, the union of the sets [latex]\\left\\{2,3,5\\right\\}[\/latex] and [latex]\\left\\{4,6\\right\\}[\/latex] is the set [latex]\\left\\{2,3,4,5,6\\right\\}.[\/latex] It is the set of all elements that belong to one <em>or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is<\/p>\n<div id=\"fs-id1165135311695\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/div>\n<div id=\"fs-id1165137641795\">\n<h3>Set-Builder Notation and Interval Notation<\/h3>\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137663670\"><strong>Set-builder notation <\/strong>is a method of specifying a set of elements that satisfy a certain condition. It takes the form [latex]\\left\\{x|\\text{ }\\text{statement about }x\\right\\}[\/latex] which is read as, \u201cthe set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.\u201d For example,<\/p>\n<div id=\"fs-id1165137543047\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\left\\{x|4\\lt x\\le 12\\right\\}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135190272\"><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example, [latex]\\left(4,12\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805770\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137423878\"><strong>Given a line graph, describe the set of values using interval notation.<\/strong><\/p>\n<ol id=\"fs-id1165134032280\" type=\"1\">\n<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\n<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\n<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\n<li>Use the union symbol, [latex]\\cup \\text{, }[\/latex]to combine all intervals into one set.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1165134342702\">\n<div id=\"fs-id1165137803670\">\n<h3>Example 5:\u00a0 Describing Sets on the Real-Number Line<\/h3>\n<p id=\"fs-id1165137592069\">Describe the intervals of values shown in <a class=\"autogenerated-content\" href=\"#Figure_01_02_004\">Figure 6<\/a> using inequality notation, set-builder notation, and interval notation.<\/p>\n<div id=\"attachment_3083\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3083\" class=\"size-full wp-image-3083\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152546\/13c74d3bc393a72d003d8fa46c769c591786bc87-1.jpeg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/><\/p>\n<p id=\"caption-attachment-3083\" class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<\/div>\n<div class=\"small\"><\/div>\n<div id=\"fs-id1165135412904\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135412904\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135412904\" 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, \u201c[latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\u201d<\/p>\n<table id=\"fs-id1165137447518\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\"><strong>Inequality<\/strong><\/td>\n<td class=\"border\">[latex]1\\le x\\le 3\\text{ }\\text{or}\\text{ }x>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>Set-builder notation<\/strong><\/td>\n<td class=\"border\">[latex]\\left\\{x|1\\le x\\le 3\\text{ }\\text{or}\\text{ }x>5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>Interval notation<\/strong><\/td>\n<td class=\"border\">[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 id=\"fs-id1165137779165\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"ti_01_02_03\">\n<div id=\"fs-id1165135175087\">\n<p id=\"fs-id1165135341412\">Given <a class=\"autogenerated-content\" href=\"#Figure_01_02_005\">Figure 7<\/a>, specify the graphed set in<\/p>\n<ol id=\"fs-id1165137595582\" type=\"a\">\n<li>words<\/li>\n<li>set-builder notation<\/li>\n<li>interval notation\n<div id=\"attachment_3084\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3084\" class=\"size-full wp-image-3084\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152753\/eba2f61d5fb32eca2dd57b7de1a1e57511a15f6a.jpeg\" alt=\"\" width=\"487\" height=\"50\" \/><\/p>\n<p id=\"caption-attachment-3084\" class=\"wp-caption-text\">Figure 7<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"small\"><\/div>\n<div id=\"ti_01_02_03\">\n<div id=\"fs-id1165135209390\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135209390\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135209390\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165135528963\" type=\"a\">\n<li>Values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3;<\/li>\n<li>[latex]\\left\\{x|x\\le -2\\text{ }\\text{or}\\text{ }-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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653855\" class=\"bc-section section\">\n<h3>Finding Domain and Range from Graphs<\/h3>\n<p id=\"fs-id1165135161404\">Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em>x<\/em>-axis. The range is the set of possible output values, which are shown on the <em>y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. This is indicated by the arrows in\u00a0<a class=\"autogenerated-content\" href=\"#Figure_01_02_006\">Figure 8<\/a>.<\/p>\n<div id=\"attachment_3085\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3085\" class=\"wp-image-3085 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152837\/aec6dfffbf1ae7c213cf7c467e97eade263a24c9.jpeg\" alt=\"\" width=\"487\" height=\"666\" \/><\/p>\n<p id=\"caption-attachment-3085\" class=\"wp-caption-text\">Figure 8<\/p>\n<\/div>\n<div id=\"Figure_01_02_006\" class=\"small\"><\/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 lower to higher 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=\"textbox examples\">\n<div id=\"fs-id1165137561401\">\n<div id=\"fs-id1165137599824\">\n<h3>Example 6:\u00a0 Finding Domain and Range from a Graph<\/h3>\n<div id=\"attachment_3086\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3086\" class=\"wp-image-3086 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14152944\/940d0e7a8d6cc139b44a525718f304d15c48916b.jpeg\" alt=\"\" width=\"487\" height=\"364\" \/><\/p>\n<p id=\"caption-attachment-3086\" class=\"wp-caption-text\">Figure 9<\/p>\n<\/div>\n<p id=\"fs-id1165135187604\">Find the domain and range of the function [latex]f[\/latex] whose graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_02_007\">Figure 9<\/a>.<\/p>\n<\/div>\n<\/div>\n<div class=\"small\"><\/div>\n<div id=\"fs-id1165137561401\">\n<div id=\"fs-id1165137575085\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137575085\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137575085\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137768165\">We can observe that the horizontal extent of the graph is \u20133 but not inclusive to 1 included, so the domain of [latex]f[\/latex] is [latex]\\left(-3,1\\right].[\/latex]<\/p>\n<p id=\"fs-id1165131968670\">The vertical extent of the graph is 0 to \u20134, so the range is [latex]\\left[-4,0\\right].[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_02_008\">Figure 10<\/a>.<\/p>\n<div id=\"Figure_01_02_008\" class=\"small\">\n<div id=\"attachment_3087\" style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3087\" class=\"size-full wp-image-3087\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153050\/258d8eebdb402305613f6b1ee6cfb9c3b0e16c0f.jpeg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"488\" height=\"364\" \/><\/p>\n<p id=\"caption-attachment-3087\" class=\"wp-caption-text\">Figure 10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_02_07\" class=\"textbox examples\">\n<div id=\"fs-id1165134182686\">\n<div id=\"fs-id1165137461643\">\n<h3>Example 7:\u00a0 Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p id=\"fs-id1165137443324\">Find the domain and range of the function [latex]f[\/latex] whose graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_02_009\">Figure\u00a0 11<\/a>.<\/p>\n<div id=\"attachment_3088\" style=\"width: 499px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3088\" class=\"size-full wp-image-3088\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153150\/e7beda4d06fbbe861ed77bf9eb06f3823c8932a7.jpeg\" 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 id=\"caption-attachment-3088\" class=\"wp-caption-text\">Figure 11<\/p>\n<\/div>\n<div id=\"Figure_01_02_009\" class=\"small\"><\/div>\n<\/div>\n<div><\/div>\n<div>(credit: modification of work by the U.S. Energy Information Administration)<a class=\"footnote\" title=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" id=\"return-footnote-38-2\" href=\"#footnote-38-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/div>\n<div id=\"fs-id1165137444311\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137444311\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137444311\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137476085\">The input quantity along the horizontal axis is \u201cyears,\u201d which we represent with the variable [latex]t[\/latex] for time. The output quantity is \u201cthousands of barrels of oil per day,\u201d which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010.[\/latex]<\/p>\n<p id=\"fs-id1165137747998\">In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545972\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"ti_01_02_04\">\n<div id=\"fs-id1165137644581\">\n<p id=\"fs-id1165137644582\">Given <a class=\"autogenerated-content\" href=\"#Figure_01_02_010\">Figure 12<\/a>, identify the domain and range using interval notation.<\/p>\n<div id=\"attachment_3091\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3091\" class=\"wp-image-3091 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153447\/d86a2e251962e6baf7039f5a05db37e4bb123d1a.jpeg\" 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 id=\"caption-attachment-3091\" class=\"wp-caption-text\">Figure 12<\/p>\n<\/div>\n<div id=\"Figure_01_02_010\" class=\"small\"><\/div>\n<\/div>\n<div id=\"fs-id1165137705252\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137705252\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137705252\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134079741\">domain = [1950,2002] range = [47,000,000,89,000,000]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434590\" class=\"precalculus qa key-takeaways\">\n<h3>Q&amp;A<\/h3>\n<p id=\"fs-id1165137812796\"><strong>Can a function\u2019s domain and range be the same?<\/strong><\/p>\n<p id=\"fs-id1165137433394\"><em>Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134384565\" class=\"bc-section section\">\n<h3>Finding Domains and Ranges of the Toolkit Functions<\/h3>\n<p id=\"fs-id1165137419914\">We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n<div id=\"attachment_3093\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3093\" class=\"size-full wp-image-3093\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14153618\/7288c31bf0e35bb4bfc8a17e8a2320aa037d56a9.jpeg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/><\/p>\n<p id=\"caption-attachment-3093\" class=\"wp-caption-text\">For the constant function [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<\/div>\n<div id=\"Figure_01_02_011\" class=\"small\"><\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_3094\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3094\" class=\"size-full wp-image-3094\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14154010\/71563b67e61e6ec6a9736686b246db9533b52ddf.jpeg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/><\/p>\n<p id=\"caption-attachment-3094\" class=\"wp-caption-text\">For the identity function [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<\/div>\n<p>\u00a0<\/p>\n<div id=\"attachment_3095\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3095\" class=\"size-full wp-image-3095\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14154302\/f38fb33b0045eeb53fa467758aad11565d334626.jpeg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/><\/p>\n<p id=\"caption-attachment-3095\" class=\"wp-caption-text\">For the absolute value function [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<\/div>\n<div id=\"Figure_01_02_013\" class=\"small\"><\/div>\n<div class=\"wp-caption-text\"><\/div>\n<div id=\"attachment_3096\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3096\" class=\"wp-image-3096 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14154936\/babe50d816f29913cd05a7b8e4c9cc7564fea57d.jpeg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/><\/p>\n<p id=\"caption-attachment-3096\" class=\"wp-caption-text\">For the quadratic function [latex]f\\left(x\\right)={x}^{2},[\/latex] the domain is all real numbers since any real number can be squared and result in a real number.\u00a0 We can also see graphically that the horizontal extent of the graph is the whole real number line. We can also see vertically, the graph does not include any negative values for the range, so the range is only nonnegative real numbers.<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<div id=\"attachment_3097\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3097\" class=\"size-full wp-image-3097\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155123\/2cf19d06d7beae28a6d80ed0898a7f43e6c32d86.jpeg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/><\/p>\n<p id=\"caption-attachment-3097\" class=\"wp-caption-text\">For the cubic function [latex]f\\left(x\\right)={x}^{3},[\/latex] the domain is all real numbers because any real number raised to the third power will result in a real number.\u00a0 Graphically we can see that 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<\/div>\n<div id=\"Figure_01_02_014\" class=\"small\"><\/div>\n<div id=\"Figure_01_02_015\" class=\"small\"><\/div>\n<div id=\"attachment_3098\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3098\" class=\"wp-image-3098 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155520\/44f1be11fdaa6b420f329bebb3a745e40531e363.jpeg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/><\/p>\n<p id=\"caption-attachment-3098\" class=\"wp-caption-text\">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. The graph above indicates all other real numbers can be included in the range. 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<\/div>\n<div id=\"Figure_01_02_016\" class=\"small\"><\/div>\n<div class=\"wp-caption-text\"><\/div>\n<div id=\"attachment_3099\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3099\" class=\"wp-image-3099 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155648\/7cac5fab3a66d73e68580db939aa2513f077076e.jpeg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/><\/p>\n<p id=\"caption-attachment-3099\" class=\"wp-caption-text\">For the reciprocal squared function [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<\/div>\n<div id=\"Figure_01_02_017\" class=\"small\"><\/div>\n<div id=\"attachment_3100\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3100\" class=\"size-full wp-image-3100\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14155812\/edda7a5a94c145e7a95e5657060c0d0b08c5ccdf.jpeg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/><\/p>\n<p id=\"caption-attachment-3100\" class=\"wp-caption-text\">For the square root function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{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[\\leftroot{1}\\uproot{2} ]{x}[\/latex] also gives us [latex]x.[\/latex]<\/p>\n<\/div>\n<div id=\"Figure_01_02_018\" class=\"small\"><\/div>\n<div id=\"attachment_3101\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3101\" class=\"size-full wp-image-3101\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160003\/5e97fc8781f0bb16f912cac0d584182c20d5ca5a.jpeg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/><\/p>\n<p id=\"caption-attachment-3101\" class=\"wp-caption-text\">For the cube root function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}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). We can see from the graph that we have not restrictions either horizontally or vertically.<\/p>\n<\/div>\n<div id=\"Figure_01_02_019\" class=\"small\"><\/div>\n<div class=\"wp-caption-text\"><\/div>\n<div id=\"fs-id1165137462732\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137611181\"><strong>Given the formula for a function, determine the domain and range.<\/strong><\/p>\n<ol id=\"fs-id1165137405229\" type=\"1\">\n<li>Exclude from the domain any input values that result in division by zero.<\/li>\n<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\n<li>Use the valid input values to determine the range of the output values.<\/li>\n<li>Look at the function graph and table values to confirm the actual function behavior.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_02_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137558723\">\n<div id=\"fs-id1165137464274\">\n<h3>Example 8:\u00a0 Finding the Domain and Range Using Toolkit Functions<\/h3>\n<p id=\"fs-id1165135613224\">Find the domain and range of [latex]f\\left(x\\right)=2{x}^{3}-x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135458670\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135458670\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135458670\" 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.\u00a0 The range can be found by graphing the function and viewing the vertical extent of the graph.\u00a0 The range of this type of function will be discussed in detail in chapter 4.<\/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=\"textbox examples\">\n<div id=\"fs-id1165137448155\">\n<div id=\"fs-id1165137661316\">\n<h3>Example 9:\u00a0 Finding the Domain and Range<\/h3>\n<p id=\"fs-id1165137419507\">Find the domain and range of [latex]f\\left(x\\right)=\\frac{2}{x+1}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137871182\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137871182\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137871182\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137855321\">We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex]\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right).[\/latex] Because the function is never zero, we exclude 0 from the range. The range is [latex]\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_02_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137574740\">\n<div id=\"fs-id1165135641583\">\n<h3>Example 10:\u00a0 Finding the Domain and Range<\/h3>\n<p id=\"fs-id1165137661054\">Find the domain and range of [latex]f\\left(x\\right)=2\\sqrt[\\leftroot{1}\\uproot{2} ]{x+4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137584342\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137584342\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137584342\" 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=\"unnumbered\" style=\"text-align: center;\">[latex]x+4\\ge 0[\/latex] when [latex]x\\ge -4[\/latex]<\/div>\n<div><\/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<h3>Analysis<\/h3>\n<p id=\"fs-id1165137437183\"><a class=\"autogenerated-content\" href=\"#Figure_01_02_020\">Figure 13<\/a> represents the function [latex]f.[\/latex]<\/p>\n<div id=\"attachment_3103\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3103\" class=\"size-full wp-image-3103\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160248\/d001600799023600e805aeb76b812fb5501ea12d-1.jpeg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/><\/p>\n<p id=\"caption-attachment-3103\" class=\"wp-caption-text\">Figure 13<\/p>\n<\/div>\n<div id=\"Figure_01_02_020\" class=\"small\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137572635\">\n<div id=\"Figure_01_02_020\" class=\"small\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137430800\" class=\"precalculus tryit\">\n<h3>Try it #7<\/h3>\n<div id=\"ti_01_02_05\">\n<div id=\"fs-id1165137475544\">\n<p id=\"fs-id1165137475545\">Find the domain and range of [latex]f\\left(x\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{2-x}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137833252\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137833252\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137833252\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137725047\">domain: [latex]\\left(-\\infty ,2\\right];[\/latex] range: [latex]\\left(-\\infty ,0\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440477\" class=\"bc-section section\">\n<div class=\"textbox examples\">\n<h3>Example 11:\u00a0 Find the Domain<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{x-1}}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q682389\">Show Solution<\/span><\/p>\n<div id=\"q682389\" class=\"hidden-answer\" style=\"display: none\">\n<p>This function involves both a fraction and an even root so we need to consider how each of these properties will effect the domain.\u00a0 First we know that the expression under the radicand, [latex]x-1,[\/latex] needs to be greater than or equal to zero.\u00a0 However, if the expression equals zero, we will get a divide by zero.\u00a0 Therefore, because the square root is in the denominator, we solve<\/p>\n<p style=\"text-align: center;\">[latex]x-1>0.[\/latex]<\/p>\n<p>This simplifies to [latex]x>1,[\/latex] so the domain is [latex]\\left(1,\\text{ }\\infty\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Graphing Piecewise-Defined Functions (Optional)<\/h3>\n<p id=\"fs-id1165137409262\">Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function [latex]f\\left(x\\right)=|x|.[\/latex] With a domain of all real numbers and a range of values greater than or equal to 0, <span class=\"no-emphasis\">absolute value<\/span> can be defined as the <span class=\"no-emphasis\">magnitude<\/span>, or <span class=\"no-emphasis\">modulus<\/span>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\n<p id=\"fs-id1165137558775\">If we input 0, or a positive value, the output is the same as the input.<\/p>\n<div id=\"fs-id1165135194329\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=x\\text{ }\\text{if}\\text{ }x\\ge 0[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137529947\">If we input a negative value, the output is the opposite of the input.<\/p>\n<div id=\"fs-id1165133112779\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-x\\text{ }\\text{if}\\text{ }x<0[\/latex][latex]\\\\[\/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.\u00a0 A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\n<p id=\"fs-id1165134042316\">We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income [latex]S[\/latex] would be [latex]0.1S[\/latex] if\u00a0[latex]S\\le 10\\text{,}000[\/latex] and [latex]1000+0.2\\left(S-10\\text{,}000\\right)[\/latex] if [latex]S>10\\text{,}000.[\/latex]<\/p>\n<div id=\"fs-id1165137531241\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165135504970\">A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:<\/p>\n<div id=\"fs-id1165137482244\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{l}\\text{formula 1 if }x\\text{ is in domain 1}\\\\ \\text{formula 2 if }x\\text{ is in domain 2}\\\\ \\text{formula 3 if }x\\text{ is in domain 3}\\end{array}[\/latex]<\/div>\n<\/div>\n<p>In piecewise notation, the absolute value function is<\/p>\n<div id=\"fs-id1165135190749\" class=\"unnumbered\" style=\"text-align: center;\">[latex]|x|=\\bigg\\{\\begin{array}{l}x\\text{ if }x\\ge 0\\\\ -x\\text{ if }x<0\\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<div id=\"fs-id1165137768426\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137823161\"><strong>Given a piecewise function, write the formula and identify the domain for each interval. <\/strong><\/p>\n<ol id=\"fs-id1165135443772\" type=\"1\">\n<li>Identify the intervals for which different rules apply.<\/li>\n<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\n<li>Use braces and if-statements to write the function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_02_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137452506\">\n<div id=\"fs-id1165135321994\">\n<h3>Example 12:\u00a0 Writing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137834905\">A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <span class=\"no-emphasis\">function<\/span> relating the number of people, [latex]n,[\/latex] to the cost, [latex]C.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137807421\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137807421\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137807421\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135331729\">Two different formulas will be needed. For <em>n<\/em>-values under 10, [latex]C=5n.[\/latex] For values of [latex]n[\/latex] that are 10 or greater, [latex]C=50.[\/latex]<\/p>\n<div id=\"fs-id1165135208951\" class=\"unnumbered\" style=\"text-align: center;\">[latex]C\\left(n\\right)=\\bigg\\{\\begin{array}{ccc}5n& \\text{if}& 0\\lt n \\lt10\\\\ 50& \\text{if}& n\\ge 10\\end{array}[\/latex]<\/div>\n<div>\n<h3>Analysis<\/h3>\n<p id=\"fs-id1165135196985\">The function is represented in <a class=\"autogenerated-content\" href=\"#Figure_01_02_021\">Figure 14<\/a>. The graph is a diagonal line from [latex]n=0[\/latex] to [latex]n=10[\/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10,[\/latex] but not all piecewise functions have this property.<\/p>\n<div id=\"attachment_3104\" style=\"width: 370px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3104\" class=\"size-full wp-image-3104\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160457\/ddd5650076d9fc0801b73b14769d868d34a4b148-1.jpeg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/><\/p>\n<p id=\"caption-attachment-3104\" class=\"wp-caption-text\">Figure 14<\/p>\n<\/div>\n<div id=\"Figure_01_02_021\" class=\"small\"><\/div>\n<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_02_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135436662\">\n<div id=\"fs-id1165135436664\">\n<h3>Example 13:\u00a0 Working with a Piecewise Function<\/h3>\n<p id=\"fs-id1165137938645\">A cell phone company uses the function below to determine the cost, [latex]C,[\/latex] in dollars for [latex]g[\/latex] gigabytes of data transfer.<\/p>\n<div id=\"fs-id1165137660470\" class=\"unnumbered\" style=\"text-align: center;\">[latex]C\\left(g\\right)=\\bigg\\{\\begin{array}{ccc}25& \\text{if}& 0 \\lt g<2\\\\ 25+10\\left(g-2\\right)& \\text{if}& g\\ge 2\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135193798\">Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.<\/p>\n<\/div>\n<div id=\"fs-id1165135177567\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135177567\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135177567\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134373545\">To find the cost of using 1.5 gigabytes of data, [latex]C\\left(1.5\\right),[\/latex] we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.<\/p>\n<div id=\"fs-id1165134300204\" style=\"text-align: center;\">[latex]C\\left(1.5\\right)=25[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135440213\">To find the cost of using 4 gigabytes of data, [latex]C\\left(4\\right),[\/latex] we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\n<div id=\"fs-id1165135383665\" style=\"text-align: center;\">[latex]C\\left(4\\right)=25+10\\left(4-2\\right)=45[\/latex]<\/div>\n<div><\/div>\n<div>\n<h3>Analysis<\/h3>\n<p id=\"fs-id1165137601265\">The function is represented in <a class=\"autogenerated-content\" href=\"#Figure_01_02_022\">Figure 15<\/a>. We can see where the function changes from a constant to a linear\u00a0 function with slope 10 at [latex]g=2.[\/latex] We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.<\/p>\n<div id=\"attachment_3105\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3105\" class=\"size-full wp-image-3105\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160604\/75d850575e89cd19aeea1fb435b9ece4313fb816.jpeg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/><\/p>\n<p id=\"caption-attachment-3105\" class=\"wp-caption-text\">Figure 15<\/p>\n<\/div>\n<div id=\"Figure_01_02_022\" class=\"\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600493\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135532516\"><strong>Given a piecewise function, sketch a graph.<\/strong><\/p>\n<ol id=\"fs-id1165137588539\" type=\"1\">\n<li>Indicate on the <em>x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\n<li>For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_02_13\" class=\"textbox examples\">\n<div id=\"fs-id1165137781618\">\n<div id=\"fs-id1165135412870\">\n<h3>Example 14:\u00a0 Graphing a Piecewise Function<\/h3>\n<p id=\"fs-id1165137838785\">Sketch a graph of the function.<\/p>\n<div id=\"fs-id1165137475346\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}{x}^{2}& \\text{if}& x\\le 1\\\\ 3& \\text{if}& 1\\lt x\\le 2\\\\ x& \\text{if}& x>2\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135487148\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135487148\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135487148\" 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><a class=\"autogenerated-content\" href=\"#Figure_01_02_023\">Figure 16<\/a> shows the three components of the piecewise function graphed on separate coordinate systems.<\/p>\n<div id=\"attachment_3106\" style=\"width: 984px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3106\" class=\"size-full wp-image-3106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160736\/8709146c78394619c275f4d39250c87b94329905.jpeg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/><\/p>\n<p id=\"caption-attachment-3106\" class=\"wp-caption-text\">Figure 16\u00a0(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]<\/p>\n<\/div>\n<p id=\"fs-id1165137676209\">Now that we have sketched each piece individually, we combine them in the same coordinate plane. See <a class=\"autogenerated-content\" href=\"#Figure_01_02_026\">Figure 17<\/a>.<\/p>\n<div id=\"Figure_01_02_026\" class=\"small\">\n<div id=\"attachment_3107\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3107\" class=\"size-full wp-image-3107\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14160824\/e20a45f4db87b9669c40bc522d232a07b8147103-1.jpeg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/><\/p>\n<p id=\"caption-attachment-3107\" class=\"wp-caption-text\">Figure 17<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Analysis<\/h3>\n<p>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] and [latex]\\left(2,\\text{ }3\\right)[\/latex] are.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762558\" class=\"precalculus tryit\">\n<h3>Try it #8<\/h3>\n<div id=\"ti_01_02_06\">\n<div id=\"fs-id1165137692562\">\n<p id=\"fs-id1165137692563\">Graph the following piecewise function.<\/p>\n<div id=\"fs-id1165137433350\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}{x}^{3}& \\text{if}& x \\lt -1\\\\ -2& \\text{if}& -1\\lt x\\lt 4\\\\ \\sqrt{x}& \\text{if}& x>4\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137784656\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137784656\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137784656\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<div><span id=\"fs-id1165134302462\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3172\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/15144941\/55b54680118bbe7214755035f529a3ba2024d244.jpeg\" alt=\"Try it 8\" width=\"487\" height=\"408\" \/><\/span><\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810682\" class=\"precalculus qa key-takeaways\">\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<div id=\"fs-id1165135190393\" class=\"precalculus media\">\n<p id=\"fs-id1165137627040\">Access these online resources for additional instruction and practice with domain and range.<\/p>\n<ul id=\"fs-id1165135189954\">\n<li><a href=\"http:\/\/openstax.org\/l\/domainsqroot\">Domain and Range of Square Root Functions<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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<ul id=\"fs-id1165135189954\">\n<li><a href=\"http:\/\/openstax.org\/l\/determinedomain\">Determining Domain and Range<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Determining Domain and Range\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/FtJRstFMdhA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"fs-id1165135189954\">\n<li><a href=\"http:\/\/openstax.org\/l\/drgraph\">Find Domain and Range Given the Graph<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Give the Domain and Range Given the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8jrkzZy04BQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"fs-id1165135189954\">\n<li><a href=\"http:\/\/openstax.org\/l\/drtable\">Find Domain and Range Given a Table<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Give the Domain and Range Given the Points in a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GPBq18fCEv4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<ul id=\"fs-id1165135189954\">\n<li><a href=\"http:\/\/openstax.org\/l\/drcoordinate\">Find Domain and Range Given Points on a Coordinate Plane<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex: Give the Domain and Range Given the Points on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xOsYVyjTM0Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134077347\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137591772\">\n<li>The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.<\/li>\n<li>The domain of a function can be determined by listing the input values of a set of ordered pairs.<\/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 for functions of real numbers 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<\/ul>\n<p>Optional:<\/p>\n<ul id=\"fs-id1165137591772\">\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<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl>\n<dt>interval notation<\/dt>\n<dd id=\"fs-id1165135190252\">a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135487256\">\n<dt>piecewise function<\/dt>\n<dd id=\"fs-id1165137452169\">a function in which more than one formula is used to define the output<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137863188\">\n<dt>set-builder notation<\/dt>\n<dd id=\"fs-id1165137863193\">a method of describing a set by a rule that all of its members obey; it takes the form[latex]\\left\\{x|\\text{ }\\text{statement about }x\\right\\}[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-38\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li> Domain and Range. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:HKkfLPmJ@6\/Domain-and-Range\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:HKkfLPmJ@6\/Domain-and-Range<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-38-1\">The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d <a href=\"http:\/\/www.the-numbers.com\/market\/genre\/Horror\">http:\/\/www.the-numbers.com\/market\/genre\/Horror<\/a>. Accessed 3\/24\/2014 <a href=\"#return-footnote-38-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-38-2\"><a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A\">http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A<\/a>. <a href=\"#return-footnote-38-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":311,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\" Domain and Range\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:HKkfLPmJ@6\/Domain-and-Range\",\"project\":\"Essential Precalcus, Part 1\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-38","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":34,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/38\/revisions"}],"predecessor-version":[{"id":3289,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/38\/revisions\/3289"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/38\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=38"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=38"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}