{"id":1792,"date":"2023-10-12T00:32:11","date_gmt":"2023-10-12T00:32:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-domain-and-range-of-functions\/"},"modified":"2025-10-24T00:15:25","modified_gmt":"2025-10-24T00:15:25","slug":"introduction-domain-and-range-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-domain-and-range-of-functions\/","title":{"raw":"Domain and Range","rendered":"Domain and Range"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li class=\"li2\"><span class=\"s1\">Find the domain of a function defined by an equation.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Write the domain and range using standard notations.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Find domain and range from a graph.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Give domain and range of\u00a0parent functions.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Graph piecewise-defined functions.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIf you\u2019re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time\u2014<em>I am Legend<\/em>, <em>Hannibal<\/em>, <em>The Ring<\/em>, <em>The Grudge<\/em>, and <em>The Conjuring<\/em>. Figure 1\u00a0shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <strong>domain<\/strong> and range. In this section we will investigate methods for determining the domain and range of functions such as these.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193523\/CNX_Precalc_Figure_01_02_0012.jpg\" alt=\"Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year\" width=\"975\" height=\"402\" \/> <b>Figure 1.<\/b> Based on data compiled by <a href=\"http:\/\/www.the-numbers.com\" target=\"_blank\" rel=\"noopener\">www.the-numbers.com<\/a>.[\/caption]\r\n<h2>Standard Notation for Defining Sets<\/h2>\r\nThere are several ways to define sets of numbers or mathematical objects. The reason we are introducing this here is because we often need to define the sets of numbers that make up the inputs and outputs of a function.\r\n\r\nHow we write sets that make up the domain and range of functions often depends on how the relation or function are defined or presented to us. \u00a0For example, we can use lists to describe the domain of functions that are given as sets of ordered pairs. If we are given an equation or graph, we might use inequalities or intervals to describe domain and range.\r\n\r\nIn this section, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three ways, inequality notation, set-builder notation, and interval notation.\r\n\r\nConsider the set [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex], which describes the behavior of [latex]x[\/latex] in set-builder notation. The braces [latex]\\{\\}[\/latex] are read as \"the set of,\" and the vertical bar [latex]|[\/latex] is read as \"such that,\" so we would read [latex]\\left\\{x|10\\le x&lt;30\\right\\}[\/latex] as \"the set of <em>x<\/em>-values such that 10 is less than or equal to [latex]x[\/latex], and [latex]x[\/latex] is less than 30.\"\r\n\r\nThe table below compares inequality notation, set-builder notation, and interval notation.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Inequality Notation<\/th>\r\n<th>Set-builder Notation<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\r\n<td>[latex]5&lt;h\\le10[\/latex]<\/td>\r\n<td>[latex]\\{h | 5 &lt; h \\le 10\\}[\/latex]<\/td>\r\n<td>[latex](5,10][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\r\n<td>[latex]5\\le h&lt;10[\/latex]<\/td>\r\n<td>[latex]\\{h | 5 \\le h &lt; 10\\}[\/latex]<\/td>\r\n<td>[latex][5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\r\n<td>[latex]5&lt;h&lt;10[\/latex]<\/td>\r\n<td>[latex]\\{h | 5 &lt; h &lt; 10\\}[\/latex]<\/td>\r\n<td>[latex](5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\r\n<td>[latex]h&lt;10[\/latex]<\/td>\r\n<td>[latex]\\{h | h &lt; 10\\}[\/latex]<\/td>\r\n<td>[latex](-\\infty,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\r\n<td>[latex]h&gt;10[\/latex]<\/td>\r\n<td>[latex]\\{h | h &gt; 10\\}[\/latex]<\/td>\r\n<td>[latex](10,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\r\n<td>All real numbers<\/td>\r\n<td>[latex]\\mathbf{R}[\/latex]<\/td>\r\n<td>[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo combine two intervals using inequality notation or set-builder notation, we use the word \"or.\" As we saw in earlier examples, we use the union symbol, [latex]\\cup [\/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\\left\\{2,3,5\\right\\}[\/latex]\u00a0and [latex]\\left\\{4,6\\right\\}[\/latex]\u00a0is the set [latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]. It is the set of all elements that belong to one <em>or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is\r\n<p style=\"text-align: center;\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\r\nThis video describes how to use interval notation to describe a set.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=hqg85P0ZMZ4\r\n\r\nThis video describes how to use Set-Builder notation to describe a set.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=rPcGeaDRnyc&amp;feature=youtu.be\r\n<div class=\"textbox\">\r\n<h3>A General Note: Set-Builder Notation and Interval Notation<\/h3>\r\nSet-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form [latex]\\left\\{x|\\text{statement about }x\\right\\}[\/latex] which is read as, \"the set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.\" For example,\r\n<p style=\"text-align: center;\">[latex]\\left\\{x|4&lt;x\\le 12\\right\\}[\/latex]<\/p>\r\n<strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,\r\n<p style=\"text-align: center;\">[latex]\\left(4,12\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a line graph, describe the set of values using interval notation.<\/h3>\r\n<ol>\r\n \t<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\r\n \t<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\r\n \t<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\r\n \t<li>Use the union symbol [latex]\\cup [\/latex] to combine all intervals into one set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Describing Sets on the Real-Number Line<\/h3>\r\nDescribe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/>\r\n[reveal-answer q=\"169160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169160\"]\r\n\r\nTo describe the values, [latex]x[\/latex], included in the intervals shown, we would say, \" [latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\"\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Inequality<\/strong><\/td>\r\n<td>[latex]1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Set-builder notation<\/strong><\/td>\r\n<td>[latex]\\left\\{x|1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x&gt;5\\right\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Interval notation<\/strong><\/td>\r\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember 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.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom18class=\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=108347&amp;theme=oea&amp;iframe_resize_id=mom18\" width=\"100%\" height=\"650\"><\/iframe>\r\n<iframe id=\"mom20\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3190&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"450\"><\/iframe>\r\n<iframe id=\"mom30\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3191&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the graph below, specify the graphed set in\r\n<ol>\r\n \t<li>words<\/li>\r\n \t<li>set-builder notation<\/li>\r\n \t<li>interval notation<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193546\/CNX_Precalc_Figure_01_02_0052.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/>\r\n\r\n[reveal-answer q=\"102737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"102737\"]\r\n\r\nWords: values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3.\r\n\r\nSet-builder notation: [latex]\\left\\{x|x\\le -2\\hspace{2mm}\\text{or}\\hspace{2mm}-1\\le x&lt;3\\right\\}[\/latex];\r\n\r\nInterval notation: [latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe table below gives\u00a0a summary of interval notation.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193529\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/>\r\n<h2>Write Domain and Range Given an Equation<\/h2>\r\nIn Functions and Function Notation, we were introduced to the concepts of <strong>domain and range<\/strong>. In this section we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193526\/CNX_Precalc_Figure_01_02_0022.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/>\r\n\r\nWe can visualize the domain as a \"holding area\" that contains \"raw materials\" for a \"function machine\" and the range as another \"holding area\" for the machine\u2019s products.\r\n\r\nWe can write the <strong>domain and range<\/strong> in <strong>interval notation<\/strong>, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write [latex]\\left(0,\\text{ }100\\right][\/latex]. We will discuss interval notation in greater detail later.\r\n\r\nLet\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.\r\n\r\nBefore we begin, let us review the conventions of interval notation:\r\n<ul>\r\n \t<li>The smallest term from the interval is written first.<\/li>\r\n \t<li>The largest term in the interval is written second, following a comma.<\/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<\/ul>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\r\nFind the domain of the following function: [latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex] .\r\n\r\n[reveal-answer q=\"202869\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"202869\"]\r\n\r\nFirst identify the input values. The input value is the first coordinate in an <strong>ordered pair<\/strong>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.\r\n<p style=\"text-align: center;\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function:\r\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"265532\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265532\"]\r\n\r\n[latex]\\left\\{-5,0,5,10,15\\right\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=72181&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3><strong>How To: Given a function written in equation form, find the domain.<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\r\n \t<li>Write the domain in interval form, if possible.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Function<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].\r\n\r\n[reveal-answer q=\"100687\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"100687\"]\r\n\r\nThe 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.\r\n\r\nIn interval form the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}[\/latex].\r\n\r\n[reveal-answer q=\"237099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"237099\"]\r\n\r\n[latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=60533&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To:\u00a0Given a function written in an equation form that includes a fraction, find the domain.<\/h3>\r\n<ol>\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] . These are the values that cannot be inputs in the function.<\/li>\r\n \t<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Function Involving a Denominator (Rational Function)<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\dfrac{x+1}{2-x}[\/latex].\r\n\r\n[reveal-answer q=\"759017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"759017\"]\r\n\r\nWhen 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].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}2-x&amp;=0 \\\\ -x&amp;=-2 \\\\ x&amp;=2 \\end{align}[\/latex]<\/p>\r\nNow, we will exclude 2 from the domain. The answers are all real numbers where [latex]x&lt;2[\/latex] or [latex]x&gt;2[\/latex]. We can use a symbol known as the union, [latex]\\cup [\/latex], to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193532\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/>\r\n\r\nIn interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&amp;feature=youtu.be\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function: [latex]f\\left(x\\right)=\\dfrac{1+4x}{2x - 1}[\/latex].\r\n\r\n[reveal-answer q=\"307426\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"307426\"]\r\n\r\n[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61836&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function written in equation form including an even root, find the domain.<\/h3>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/li>\r\n \t<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Function with an Even Root<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].\r\n\r\n[reveal-answer q=\"722013\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"722013\"]\r\n\r\nWhen there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.\r\n\r\nSet the radicand greater than or equal to zero and solve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}7-x&amp;\\ge 0 \\\\ -x&amp;\\ge -7 \\\\ x&amp;\\le 7 \\end{align}[\/latex]<\/p>\r\nNow, 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].\r\n\r\n[reveal-answer q=\"643325\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"643325\"][latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=30831&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92940&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can there be functions in which the domain and range do not intersect at all?<\/strong>\r\n\r\n<em>Yes. For example, the function [latex]f\\left(x\\right)=-\\frac{1}{\\sqrt{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em>\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\nWhen you are defining the domain of a function, it can help to graph it, especially when you have a rational or a function with an even root.\r\n\r\nFirst, determine the domain restrictions for the following functions, then graph each one to check whether your domain agrees with the graph.\r\n<ol>\r\n \t<li>[latex]f(x) = \\sqrt{2x-4}+5[\/latex]<\/li>\r\n \t<li>[latex]g(x) = \\dfrac{2x+4}{x-1}[\/latex]<\/li>\r\n<\/ol>\r\nNext, use an online graphing tool to evaluate your function at the domain restriction you found. What function value does a graphing calculator give you?\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"mceTemp\"><\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the formula for a function, determine the domain and range.<\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain and Range Using PARENT Functions<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=2{x}^{3}-x[\/latex].\r\n\r\n[reveal-answer q=\"618770\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"618770\"]\r\n\r\nThere are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] and the range is also [latex]\\left(-\\infty ,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain and Range<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=\\dfrac{2}{x+1}[\/latex].\r\n\r\n[reveal-answer q=\"71516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"71516\"]\r\n\r\nWe cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex]\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right)[\/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain and Range<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].\r\n\r\n[reveal-answer q=\"605324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"605324\"]\r\n\r\nWe cannot take the square root of a negative number, so the value inside the radical must be nonnegative.\r\n\r\n[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]\r\n\r\nThe domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].\r\n\r\nWe 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].\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph below represents the function [latex]f[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193624\/CNX_Precalc_Figure_01_02_0202.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=-\\sqrt{2-x}[\/latex].\r\n\r\n[reveal-answer q=\"336525\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"336525\"]\r\n\r\nDomain: [latex]\\left(-\\infty ,2\\right][\/latex] \u00a0 Range: [latex]\\left(-\\infty ,0\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Determine Domain and Range from a Graph<\/h3>\r\nAnother way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[\/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[\/latex]-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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193549\/CNX_Precalc_Figure_01_02_0062.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/>\r\n\r\nWe can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Domain and Range from a Graph<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193551\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/>\r\n[reveal-answer q=\"495787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495787\"]\r\n\r\nWe can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193553\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/>\r\n\r\nThe vertical extent of the graph is 0 to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=QAxZEelInJc\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Domain and Range from a Graph of Oil Production<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"489\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193556\/CNX_Precalc_Figure_01_02_0092.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/> (credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)[\/caption]\r\n\r\n[reveal-answer q=\"834467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834467\"]\r\n\r\nThe input quantity along the horizontal axis is \"years,\" which we represent with the variable [latex]t[\/latex] for time. The output quantity is \"thousands of barrels of oil per day,\" which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].\r\n\r\nIn interval notation, the domain is [latex][1973, 2008][\/latex], and the range is about [latex][180, 2010][\/latex]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the graph, identify the domain and range using interval notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193558\/CNX_Precalc_Figure_01_02_0102.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/>\r\n[reveal-answer q=\"186149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186149\"]\r\n\r\nDomain = [latex][1950, 2002][\/latex]\u00a0 \u00a0Range = [latex][47,000,000, 89,000,000][\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3765&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can a function\u2019s domain and range be the same?<\/strong>\r\n\r\n<em>Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em>\r\n\r\n<\/div>\r\n<h2>Domain and Range of Parent Functions<\/h2>\r\nWe will now return to our set of\u00a0parent functions to determine the domain and range of each.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193601\/CNX_Precalc_Figure_01_02_0112.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/> For the <strong>constant function<\/strong> [latex]f\\left(x\\right)=c[\/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[\/latex], so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right][\/latex], the interval that both begins and ends with [latex]c[\/latex].[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193604\/CNX_Precalc_Figure_01_02_0122.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/> For the <strong>identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193606\/CNX_Precalc_Figure_01_02_0132.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/> For the <strong>absolute value function<\/strong> [latex]f\\left(x\\right)=|x|[\/latex], there is no restriction on [latex]x[\/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193609\/CNX_Precalc_Figure_01_02_0142.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/> For the <strong>quadratic function<\/strong> [latex]f\\left(x\\right)={x}^{2}[\/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193611\/CNX_Precalc_Figure_01_02_0152.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/> For the <strong>cubic function<\/strong> [latex]f\\left(x\\right)={x}^{3}[\/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193614\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/> For the <strong>reciprocal function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193617\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/> For the <strong>reciprocal squared function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193620\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/> For the <strong>square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex].[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193622\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/> For the <strong>cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).[\/caption]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47481&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"750\"><\/iframe>\r\n<iframe id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47483&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"650\"><\/iframe>\r\n<iframe id=\"mom9\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47484&amp;theme=oea&amp;iframe_resize_id=mom9\" width=\"100%\" height=\"550\"><\/iframe>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47487&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Piecewise-Defined Functions<\/h2>\r\nSometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the parent functions, we introduced the absolute value function [latex]f\\left(x\\right)=|x|[\/latex]. With a domain of all real numbers and a range of values greater than or equal to 0, <strong>absolute value<\/strong> can be defined as the <strong>magnitude<\/strong>, or <strong>modulus<\/strong>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.\r\n\r\nIf we input 0, or a positive value, the output is the same as the input.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=x\\text{ if }x\\ge 0[\/latex]<\/p>\r\nIf we input a negative value, the output is the opposite of the input.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-x\\text{ if }x&lt;0[\/latex]<\/p>\r\nBecause this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output over different pieces of the domain.\r\n\r\nWe use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \"boundaries.\" For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to [latex]$10,000[\/latex] are taxed at [latex]10%[\/latex], and any additional income is taxed at [latex]20%[\/latex]. The tax on a total income, [latex] S[\/latex] , would be\u00a0[latex] 0.1S[\/latex] if [latex]{S}\\le$10,000[\/latex] \u00a0and [latex]1000 + 0.2 (S - $10,000)[\/latex] ,\u00a0if [latex] S&gt; $10,000[\/latex] .\r\n<div class=\"textbox\">\r\n<h3>A General Note: Piecewise Functions<\/h3>\r\nA piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:\r\n<p style=\"text-align: center;\">[latex] f\\left(x\\right)=\\begin{cases}\\text{formula 1 if x is in domain 1}\\\\ \\text{formula 2 if x is in domain 2}\\\\ \\text{formula 3 if x is in domain 3}\\end{cases} [\/latex]<\/p>\r\nIn piecewise notation, the absolute value function is\r\n<p style=\"text-align: center;\">[latex]|x|=\\begin{cases}\\begin{align}x&amp;\\text{ if }x\\ge 0\\\\ -x&amp;\\text{ if }x&lt;0\\end{align}\\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To:\u00a0Given a piecewise function, write the formula and identify the domain for each interval.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the intervals for which different rules apply.<\/li>\r\n \t<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\r\n \t<li>Use braces and if-statements to write the function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Piecewise Function<\/h3>\r\nA museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <strong>function<\/strong> relating the number of people, [latex]n[\/latex], to the cost, [latex]C[\/latex].\r\n\r\n[reveal-answer q=\"668439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668439\"]\r\n\r\nTwo different formulas will be needed. For [latex]n[\/latex]-values under 10, [latex]C=5n[\/latex]. For values of [latex]n[\/latex] that are 10 or greater, [latex]C=50[\/latex].\r\n<p style=\"text-align: center;\">[latex]C(n)=\\begin{cases}\\begin{align}{5n}&amp;\\hspace{2mm}\\text{if}\\hspace{2mm}{0}&lt;{n}&lt;{10}\\\\ 50&amp;\\hspace{2mm}\\text{if}\\hspace{2mm}{n}\\ge 10\\end{align}\\end{cases}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph is a diagonal line from [latex]n=0[\/latex] to [latex]n=10[\/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10[\/latex], but not all piecewise functions have this property.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193627\/CNX_Precalc_Figure_01_02_0212.jpg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111812&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93008&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Working with a Piecewise Function<\/h3>\r\nA cell phone company uses the function below to determine the cost, [latex]C[\/latex], in dollars for [latex]g[\/latex] gigabytes of data transfer.\r\n<p style=\"text-align: center;\">[latex]C\\left(g\\right)=\\begin{cases}\\begin{align}{25} \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ 0 }&lt;{ g }&lt;{ 2 }\\\\ { 25+10 }\\left(g - 2\\right) \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ g}\\ge{ 2 }\\end{align}\\end{cases}[\/latex]<\/p>\r\nFind the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.\r\n\r\n[reveal-answer q=\"220698\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"220698\"]\r\n\r\nTo find the cost of using 1.5 gigabytes of data, [latex]C(1.5)[\/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.\r\n<p style=\"text-align: center;\">[latex]C(1.5) = $25[\/latex]<\/p>\r\nTo find the cost of using 4 gigabytes of data, [latex]C(4)[\/latex], we see that our input of 4 is greater than 2, so we use the second formula.\r\n<p style=\"text-align: center;\">[latex]C(4)=25 + 10( 4-2) =$45[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can see where the function changes from a constant to a shifted and stretched identity at [latex]g=2[\/latex]. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193630\/CNX_Precalc_Figure_01_02_0222.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1657&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To:\u00a0Given a piecewise function, sketch a graph.<\/h3>\r\n<ol>\r\n \t<li>Indicate on the [latex]x[\/latex]-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\nhttps:\/\/www.youtube.com\/watch?v=B1jfpiI-QQ8&amp;feature=youtu.be\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Piecewise Function<\/h3>\r\nSketch a graph of the function.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}{ x }^{2} \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ x }\\le{ 1 }\\\\ { 3 } \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm} { 1 }&amp;lt{ x }\\le 2\\\\ { x } \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ x }&amp;gt{ 2 }\\end{align}\\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"375071\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"375071\"]\r\nEach of the component functions is from our library of\u00a0parent 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.\r\n\r\nBelow are\u00a0the three components of the piecewise function graphed on separate coordinate systems.\r\n\r\n(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]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193632\/CNX_Precalc_Figure_01_02_023abc2.jpg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/>\r\n\r\nNow that we have sketched each piece individually, we combine them in the same coordinate plane.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193635\/CNX_Precalc_Figure_01_02_0262.jpg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/>\r\n<h4>Analysis of the Solution<\/h4>\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]\u00a0and [latex]\\left(2,3\\right)[\/latex] are.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph the following piecewise function.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}{ x}^{3} \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ x }&amp;lt{-1 }\\\\ { -2 } \\hspace{2mm}&amp;\\text{ if } \\hspace{2mm}{ -1 }&amp;lt{ x }&amp;lt{ 4 }\\\\ \\sqrt{x} \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ x }&amp;gt{ 4 }\\end{align}\\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"432812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"432812\"]\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/courses.candelalearning.com\/osprecalc\/wp-content\/uploads\/sites\/402\/2015\/06\/CNX_Precalc_Figure_01_02_0272.jpg\" alt=\"Graph of f(x).\" width=\"487\" height=\"408\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32883&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nYou can use an online graphing tool to graph piecewise defined functions. Watch this tutorial video to learn how.\r\n\r\nhttps:\/\/youtu.be\/vmqiJV1FqwU\r\n\r\nGraph the following piecewise function with an online graphing tool.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}{ x}^{3} \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ x }&amp;lt{-1 }\\\\ { -2 } \\hspace{2mm}&amp;\\text{ if } \\hspace{2mm}{ -1 }&amp;lt{ x }&amp;lt{ 4 }\\\\ \\sqrt{x} \\hspace{2mm}&amp;\\text{ if }\\hspace{2mm}{ x }&amp;gt{ 4 }\\end{align}\\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=32883&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q&amp;A<\/h3>\r\n<strong>Can more than one formula from a piecewise function be applied to a value in the domain?<\/strong>\r\n\r\n<em>No. Each value corresponds to one equation in a piecewise formula.<\/em>\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.<\/li>\r\n \t<li>The domain of a function can be determined by listing the input values of a set of ordered pairs.<\/li>\r\n \t<li>The domain of a function can also be determined by identifying the input values of a function written as an equation.<\/li>\r\n \t<li>Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.<\/li>\r\n \t<li>For many functions, the domain and range can be determined from a graph.<\/li>\r\n \t<li>An understanding of\u00a0parent functions can be used to find the domain and range of related functions.<\/li>\r\n \t<li>A piecewise function is described by more than one formula.<\/li>\r\n \t<li>A piecewise function can be graphed using each algebraic formula on its assigned subdomain.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135445751\" class=\"definition\">\r\n \t<dt><strong>interval notation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135190252\">a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135487256\" class=\"definition\">\r\n \t<dt><strong>piecewise function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137452169\">a function in which more than one formula is used to define the output<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137863188\" class=\"definition\">\r\n \t<dt><strong>set-builder notation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137863193\">a method of describing a set by a rule that all of its members obey; it takes the form [latex]\\left\\{x|\\text{statement about }x\\right\\}[\/latex]<\/dd>\r\n<\/dl>\r\n<img class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/>\r\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and complete Homework 1.4 - Domain and Range<\/span><\/h3>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li class=\"li2\"><span class=\"s1\">Find the domain of a function defined by an equation.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Write the domain and range using standard notations.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Find domain and range from a graph.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Give domain and range of\u00a0parent functions.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Graph piecewise-defined functions.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>If you\u2019re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time\u2014<em>I am Legend<\/em>, <em>Hannibal<\/em>, <em>The Ring<\/em>, <em>The Grudge<\/em>, and <em>The Conjuring<\/em>. Figure 1\u00a0shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <strong>domain<\/strong> and range. In this section we will investigate methods for determining the domain and range of functions such as these.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193523\/CNX_Precalc_Figure_01_02_0012.jpg\" alt=\"Two graphs where the first graph is of the Top-Five Grossing Horror Movies for years 2000-2003 and Market Share of Horror Movies by Year\" width=\"975\" height=\"402\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> Based on data compiled by <a href=\"http:\/\/www.the-numbers.com\" target=\"_blank\" rel=\"noopener\">www.the-numbers.com<\/a>.<\/p>\n<\/div>\n<h2>Standard Notation for Defining Sets<\/h2>\n<p>There are several ways to define sets of numbers or mathematical objects. The reason we are introducing this here is because we often need to define the sets of numbers that make up the inputs and outputs of a function.<\/p>\n<p>How we write sets that make up the domain and range of functions often depends on how the relation or function are defined or presented to us. \u00a0For example, we can use lists to describe the domain of functions that are given as sets of ordered pairs. If we are given an equation or graph, we might use inequalities or intervals to describe domain and range.<\/p>\n<p>In this section, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three ways, inequality notation, set-builder notation, and interval notation.<\/p>\n<p>Consider the set [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex], which describes the behavior of [latex]x[\/latex] in set-builder notation. The braces [latex]\\{\\}[\/latex] are read as &#8220;the set of,&#8221; and the vertical bar [latex]|[\/latex] is read as &#8220;such that,&#8221; so we would read [latex]\\left\\{x|10\\le x<30\\right\\}[\/latex] as &#8220;the set of <em>x<\/em>-values such that 10 is less than or equal to [latex]x[\/latex], and [latex]x[\/latex] is less than 30.&#8221;<\/p>\n<p>The table below compares inequality notation, set-builder notation, and interval notation.<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th>Inequality Notation<\/th>\n<th>Set-builder Notation<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\n<td>[latex]5<h\\le10[\/latex]<\/td>\n<td>[latex]\\{h | 5 < h \\le 10\\}[\/latex]<\/td>\n<td>[latex](5,10][\/latex]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\n<td>[latex]5\\le h<10[\/latex]<\/td>\n<td>[latex]\\{h | 5 \\le h < 10\\}[\/latex]<\/td>\n<td>[latex][5,10)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\n<td>[latex]5<h<10[\/latex]<\/td>\n<td>[latex]\\{h | 5 < h < 10\\}[\/latex]<\/td>\n<td>[latex](5,10)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\n<td>[latex]h<10[\/latex]<\/td>\n<td>[latex]\\{h | h < 10\\}[\/latex]<\/td>\n<td>[latex](-\\infty,10)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\n<td>[latex]h>10[\/latex]<\/td>\n<td>[latex]\\{h | h > 10\\}[\/latex]<\/td>\n<td>[latex](10,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\n<td>All real numbers<\/td>\n<td>[latex]\\mathbf{R}[\/latex]<\/td>\n<td>[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To combine two intervals using inequality notation or set-builder notation, we use the word &#8220;or.&#8221; As we saw in earlier examples, we use the union symbol, [latex]\\cup[\/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\\left\\{2,3,5\\right\\}[\/latex]\u00a0and [latex]\\left\\{4,6\\right\\}[\/latex]\u00a0is the set [latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]. It is the set of all elements that belong to one <em>or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{x|\\text{ }|x|\\ge 3\\right\\}=\\left(-\\infty ,-3\\right]\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\n<p>This video describes how to use interval notation to describe a set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hqg85P0ZMZ4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This video describes how to use Set-Builder notation to describe a set.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Set-Builder Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rPcGeaDRnyc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Set-Builder Notation and Interval Notation<\/h3>\n<p>Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form [latex]\\left\\{x|\\text{statement about }x\\right\\}[\/latex] which is read as, &#8220;the set of all [latex]x[\/latex] such that the statement about [latex]x[\/latex] is true.&#8221; For example,<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{x|4<x\\le 12\\right\\}[\/latex]<\/p>\n<p><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,<\/p>\n<p style=\"text-align: center;\">[latex]\\left(4,12\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a line graph, describe the set of values using interval notation.<\/h3>\n<ol>\n<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\n<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\n<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\n<li>Use the union symbol [latex]\\cup[\/latex] to combine all intervals into one set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Describing Sets on the Real-Number Line<\/h3>\n<p>Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q169160\">Show Solution<\/span><\/p>\n<div id=\"q169160\" class=\"hidden-answer\" style=\"display: none\">\n<p>To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, &#8221; [latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.&#8221;<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Inequality<\/strong><\/td>\n<td>[latex]1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Set-builder notation<\/strong><\/td>\n<td>[latex]\\left\\{x|1\\le x\\le 3\\hspace{2mm}\\text{or}\\hspace{2mm}x>5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Interval notation<\/strong><\/td>\n<td>[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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<p><iframe loading=\"lazy\" id=\"mom18class=\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=108347&amp;theme=oea&amp;iframe_resize_id=mom18\" width=\"100%\" height=\"650\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom20\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3190&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"450\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom30\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3191&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph below, specify the graphed set in<\/p>\n<ol>\n<li>words<\/li>\n<li>set-builder notation<\/li>\n<li>interval notation<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193546\/CNX_Precalc_Figure_01_02_0052.jpg\" alt=\"Line graph of -2&lt;=x, -1&lt;=x&lt;3.\" width=\"487\" height=\"50\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q102737\">Show Solution<\/span><\/p>\n<div id=\"q102737\" class=\"hidden-answer\" style=\"display: none\">\n<p>Words: values that are less than or equal to \u20132, or values that are greater than or equal to \u20131 and less than 3.<\/p>\n<p>Set-builder notation: [latex]\\left\\{x|x\\le -2\\hspace{2mm}\\text{or}\\hspace{2mm}-1\\le x<3\\right\\}[\/latex];\n\nInterval notation: [latex]\\left(-\\infty ,-2\\right]\\cup \\left[-1,3\\right)[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<p>The table below gives\u00a0a summary of interval notation.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193529\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/><\/p>\n<h2>Write Domain and Range Given an Equation<\/h2>\n<p>In Functions and Function Notation, we were introduced to the concepts of <strong>domain and range<\/strong>. In this section we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193526\/CNX_Precalc_Figure_01_02_0022.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/><\/p>\n<p>We can visualize the domain as a &#8220;holding area&#8221; that contains &#8220;raw materials&#8221; for a &#8220;function machine&#8221; and the range as another &#8220;holding area&#8221; for the machine\u2019s products.<\/p>\n<p>We can write the <strong>domain and range<\/strong> in <strong>interval notation<\/strong>, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write [latex]\\left(0,\\text{ }100\\right][\/latex]. We will discuss interval notation in greater detail later.<\/p>\n<p>Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\n<p>Before we begin, let us review the conventions of interval notation:<\/p>\n<ul>\n<li>The smallest term from the interval is written first.<\/li>\n<li>The largest term in the interval is written second, following a comma.<\/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<\/ul>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\n<p>Find the domain of the following function: [latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q202869\">Show Solution<\/span><\/p>\n<div id=\"q202869\" class=\"hidden-answer\" style=\"display: none\">\n<p>First identify the input values. The input value is the first coordinate in an <strong>ordered pair<\/strong>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function:<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265532\">Show Solution<\/span><\/p>\n<div id=\"q265532\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{-5,0,5,10,15\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=72181&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3><strong>How To: Given a function written in equation form, find the domain.<\/strong><\/h3>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\n<li>Write the domain in interval form, if possible.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Function<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q100687\">Show Solution<\/span><\/p>\n<div id=\"q100687\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>In interval form the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q237099\">Show Solution<\/span><\/p>\n<div id=\"q237099\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=60533&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To:\u00a0Given a function written in an equation form that includes a fraction, find the domain.<\/h3>\n<ol>\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] . These are the values that cannot be inputs in the function.<\/li>\n<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Function Involving a Denominator (Rational Function)<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\dfrac{x+1}{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q759017\">Show Solution<\/span><\/p>\n<div id=\"q759017\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p style=\"text-align: center;\">[latex]\\begin{align}2-x&=0 \\\\ -x&=-2 \\\\ x&=2 \\end{align}[\/latex]<\/p>\n<p>Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[\/latex] or [latex]x>2[\/latex]. We can use a symbol known as the union, [latex]\\cup[\/latex], to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193532\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/><\/p>\n<p>In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  The Domain of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function: [latex]f\\left(x\\right)=\\dfrac{1+4x}{2x - 1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q307426\">Show Solution<\/span><\/p>\n<div id=\"q307426\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61836&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function written in equation form including an even root, find the domain.<\/h3>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/li>\n<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Function with an Even Root<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q722013\">Show Solution<\/span><\/p>\n<div id=\"q722013\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}7-x&\\ge 0 \\\\ -x&\\ge -7 \\\\ x&\\le 7 \\end{align}[\/latex]<\/p>\n<p>Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[\/latex], or [latex]\\left(-\\infty ,7\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Domain and Range of Square Root Functions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q643325\">Show Solution<\/span><\/p>\n<div id=\"q643325\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=30831&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92940&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\n<p><em>Yes. For example, the function [latex]f\\left(x\\right)=-\\frac{1}{\\sqrt{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\n<\/div>\n<h2><\/h2>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>When you are defining the domain of a function, it can help to graph it, especially when you have a rational or a function with an even root.<\/p>\n<p>First, determine the domain restrictions for the following functions, then graph each one to check whether your domain agrees with the graph.<\/p>\n<ol>\n<li>[latex]f(x) = \\sqrt{2x-4}+5[\/latex]<\/li>\n<li>[latex]g(x) = \\dfrac{2x+4}{x-1}[\/latex]<\/li>\n<\/ol>\n<p>Next, use an online graphing tool to evaluate your function at the domain restriction you found. What function value does a graphing calculator give you?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"mceTemp\"><\/div>\n<div class=\"textbox\">\n<h3>How To: Given the formula for a function, determine the domain and range.<\/h3>\n<ol>\n<li>Exclude from the domain any input values that result in division by zero.<\/li>\n<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\n<li>Use the valid input values to determine the range of the output values.<\/li>\n<li>Look at the function graph and table values to confirm the actual function behavior.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain and Range Using PARENT Functions<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=2{x}^{3}-x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q618770\">Show Solution<\/span><\/p>\n<div id=\"q618770\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.<\/p>\n<p>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 class=\"textbox exercises\">\n<h3>Example: Finding the Domain and Range<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=\\dfrac{2}{x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q71516\">Show Solution<\/span><\/p>\n<div id=\"q71516\" class=\"hidden-answer\" style=\"display: none\">\n<p>We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex]\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right)[\/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain and Range<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q605324\">Show Solution<\/span><\/p>\n<div id=\"q605324\" class=\"hidden-answer\" style=\"display: none\">\n<p>We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\n<p>[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/p>\n<p>The domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].<\/p>\n<p>We then find the range. We know that [latex]f\\left(-4\\right)=0[\/latex], and the function value increases as [latex]x[\/latex] increases without any upper limit. We conclude that the range of [latex]f[\/latex] is [latex]\\left[0,\\infty \\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph below represents the function [latex]f[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193624\/CNX_Precalc_Figure_01_02_0202.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=-\\sqrt{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q336525\">Show Solution<\/span><\/p>\n<div id=\"q336525\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain: [latex]\\left(-\\infty ,2\\right][\/latex] \u00a0 Range: [latex]\\left(-\\infty ,0\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Determine Domain and Range from a Graph<\/h3>\n<p>Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[\/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[\/latex]-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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193549\/CNX_Precalc_Figure_01_02_0062.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/><\/p>\n<p>We can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Domain and Range from a Graph<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193551\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495787\">Show Solution<\/span><\/p>\n<div id=\"q495787\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193553\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/><\/p>\n<p>The vertical extent of the graph is 0 to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 1:  Determine the Domain and Range of the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex].<\/p>\n<div style=\"width: 499px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193556\/CNX_Precalc_Figure_01_02_0092.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\">(credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834467\">Show Solution<\/span><\/p>\n<div id=\"q834467\" class=\"hidden-answer\" style=\"display: none\">\n<p>The input quantity along the horizontal axis is &#8220;years,&#8221; which we represent with the variable [latex]t[\/latex] for time. The output quantity is &#8220;thousands of barrels of oil per day,&#8221; which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].<\/p>\n<p>In interval notation, the domain is [latex][1973, 2008][\/latex], and the range is about [latex][180, 2010][\/latex]. 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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph, identify the domain and range using interval notation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193558\/CNX_Precalc_Figure_01_02_0102.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186149\">Show Solution<\/span><\/p>\n<div id=\"q186149\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex][1950, 2002][\/latex]\u00a0 \u00a0Range = [latex][47,000,000, 89,000,000][\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3765&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can a function\u2019s domain and range be the same?<\/strong><\/p>\n<p><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<h2>Domain and Range of Parent Functions<\/h2>\n<p>We will now return to our set of\u00a0parent functions to determine the domain and range of each.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193601\/CNX_Precalc_Figure_01_02_0112.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>constant function<\/strong> [latex]f\\left(x\\right)=c[\/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[\/latex], so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right][\/latex], the interval that both begins and ends with [latex]c[\/latex].<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193604\/CNX_Precalc_Figure_01_02_0122.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193606\/CNX_Precalc_Figure_01_02_0132.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>absolute value function<\/strong> [latex]f\\left(x\\right)=|x|[\/latex], there is no restriction on [latex]x[\/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193609\/CNX_Precalc_Figure_01_02_0142.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>quadratic function<\/strong> [latex]f\\left(x\\right)={x}^{2}[\/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193611\/CNX_Precalc_Figure_01_02_0152.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>cubic function<\/strong> [latex]f\\left(x\\right)={x}^{3}[\/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193614\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>reciprocal function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193617\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>reciprocal squared function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193620\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex].<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193622\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47481&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"750\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47483&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"650\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom9\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47484&amp;theme=oea&amp;iframe_resize_id=mom9\" width=\"100%\" height=\"550\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47487&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<h2>Piecewise-Defined Functions<\/h2>\n<p>Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the parent functions, we introduced the absolute value function [latex]f\\left(x\\right)=|x|[\/latex]. With a domain of all real numbers and a range of values greater than or equal to 0, <strong>absolute value<\/strong> can be defined as the <strong>magnitude<\/strong>, or <strong>modulus<\/strong>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\n<p>If we input 0, or a positive value, the output is the same as the input.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=x\\text{ if }x\\ge 0[\/latex]<\/p>\n<p>If we input a negative value, the output is the opposite of the input.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-x\\text{ if }x<0[\/latex]<\/p>\n<p>Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\n<p>We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain &#8220;boundaries.&#8221; For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to [latex]$10,000[\/latex] are taxed at [latex]10%[\/latex], and any additional income is taxed at [latex]20%[\/latex]. The tax on a total income, [latex]S[\/latex] , would be\u00a0[latex]0.1S[\/latex] if [latex]{S}\\le$10,000[\/latex] \u00a0and [latex]1000 + 0.2 (S - $10,000)[\/latex] ,\u00a0if [latex]S> $10,000[\/latex] .<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Piecewise Functions<\/h3>\n<p>A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\text{formula 1 if x is in domain 1}\\\\ \\text{formula 2 if x is in domain 2}\\\\ \\text{formula 3 if x is in domain 3}\\end{cases}[\/latex]<\/p>\n<p>In piecewise notation, the absolute value function is<\/p>\n<p style=\"text-align: center;\">[latex]|x|=\\begin{cases}\\begin{align}x&\\text{ if }x\\ge 0\\\\ -x&\\text{ if }x<0\\end{align}\\end{cases}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To:\u00a0Given a piecewise function, write the formula and identify the domain for each interval.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the intervals for which different rules apply.<\/li>\n<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\n<li>Use braces and if-statements to write the function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Piecewise Function<\/h3>\n<p>A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <strong>function<\/strong> relating the number of people, [latex]n[\/latex], to the cost, [latex]C[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q668439\">Show Solution<\/span><\/p>\n<div id=\"q668439\" class=\"hidden-answer\" style=\"display: none\">\n<p>Two different formulas will be needed. For [latex]n[\/latex]-values under 10, [latex]C=5n[\/latex]. For values of [latex]n[\/latex] that are 10 or greater, [latex]C=50[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]C(n)=\\begin{cases}\\begin{align}{5n}&\\hspace{2mm}\\text{if}\\hspace{2mm}{0}<{n}<{10}\\\\ 50&\\hspace{2mm}\\text{if}\\hspace{2mm}{n}\\ge 10\\end{align}\\end{cases}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193627\/CNX_Precalc_Figure_01_02_0212.jpg\" alt=\"Graph of C(n).\" width=\"360\" height=\"294\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111812&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93008&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Working with a Piecewise Function<\/h3>\n<p>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<p style=\"text-align: center;\">[latex]C\\left(g\\right)=\\begin{cases}\\begin{align}{25} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ 0 }<{ g }<{ 2 }\\\\ { 25+10 }\\left(g - 2\\right) \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ g}\\ge{ 2 }\\end{align}\\end{cases}[\/latex]<\/p>\n<p>Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q220698\">Show Solution<\/span><\/p>\n<div id=\"q220698\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the cost of using 1.5 gigabytes of data, [latex]C(1.5)[\/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<p style=\"text-align: center;\">[latex]C(1.5) = $25[\/latex]<\/p>\n<p>To find the cost of using 4 gigabytes of data, [latex]C(4)[\/latex], we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\n<p style=\"text-align: center;\">[latex]C(4)=25 + 10( 4-2) =$45[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can see where the function changes from a constant to a shifted and stretched identity at [latex]g=2[\/latex]. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193630\/CNX_Precalc_Figure_01_02_0222.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1657&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To:\u00a0Given a piecewise function, sketch a graph.<\/h3>\n<ol>\n<li>Indicate on the [latex]x[\/latex]-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<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex 2:  Graph a Piecewise Defined Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B1jfpiI-QQ8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Piecewise Function<\/h3>\n<p>Sketch a graph of the function.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}{ x }^{2} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }\\le{ 1 }\\\\ { 3 } \\hspace{2mm}&\\text{ if }\\hspace{2mm} { 1 }&lt{ x }\\le 2\\\\ { x } \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }&gt{ 2 }\\end{align}\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q375071\">Show Solution<\/span><\/p>\n<div id=\"q375071\" class=\"hidden-answer\" style=\"display: none\">\nEach of the component functions is from our library of\u00a0parent 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>Below are\u00a0the three components of the piecewise function graphed on separate coordinate systems.<\/p>\n<p>(a) [latex]f\\left(x\\right)={x}^{2}\\text{ if }x\\le 1[\/latex]; (b) [latex]f\\left(x\\right)=3\\text{ if 1< }x\\le 2[\/latex]; (c) [latex]f\\left(x\\right)=x\\text{ if }x>2[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193632\/CNX_Precalc_Figure_01_02_023abc2.jpg\" alt=\"Graph of each part of the piece-wise function f(x)\" width=\"974\" height=\"327\" \/><\/p>\n<p>Now that we have sketched each piece individually, we combine them in the same coordinate plane.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193635\/CNX_Precalc_Figure_01_02_0262.jpg\" alt=\"Graph of the entire function.\" width=\"487\" height=\"333\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\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]\u00a0and [latex]\\left(2,3\\right)[\/latex] are.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph the following piecewise function.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}{ x}^{3} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }&lt{-1 }\\\\ { -2 } \\hspace{2mm}&\\text{ if } \\hspace{2mm}{ -1 }&lt{ x }&lt{ 4 }\\\\ \\sqrt{x} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }&gt{ 4 }\\end{align}\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q432812\">Show Solution<\/span><\/p>\n<div id=\"q432812\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/courses.candelalearning.com\/osprecalc\/wp-content\/uploads\/sites\/402\/2015\/06\/CNX_Precalc_Figure_01_02_0272.jpg\" alt=\"Graph of f(x).\" width=\"487\" height=\"408\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32883&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>You can use an online graphing tool to graph piecewise defined functions. Watch this tutorial video to learn how.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Piecewise Functions in Desmos\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vmqiJV1FqwU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Graph the following piecewise function with an online graphing tool.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\begin{align}{ x}^{3} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }&lt{-1 }\\\\ { -2 } \\hspace{2mm}&\\text{ if } \\hspace{2mm}{ -1 }&lt{ x }&lt{ 4 }\\\\ \\sqrt{x} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }&gt{ 4 }\\end{align}\\end{cases}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=32883&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q&amp;A<\/h3>\n<p><strong>Can more than one formula from a piecewise function be applied to a value in the domain?<\/strong><\/p>\n<p><em>No. Each value corresponds to one equation in a piecewise formula.<\/em><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.<\/li>\n<li>The domain of a function can be determined by listing the input values of a set of ordered pairs.<\/li>\n<li>The domain of a function can also be determined by identifying the input values of a function written as an equation.<\/li>\n<li>Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.<\/li>\n<li>For many functions, the domain and range can be determined from a graph.<\/li>\n<li>An understanding of\u00a0parent functions can be used to find the domain and range of related functions.<\/li>\n<li>A piecewise function is described by more than one formula.<\/li>\n<li>A piecewise function can be graphed using each algebraic formula on its assigned subdomain.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135445751\" class=\"definition\">\n<dt><strong>interval notation<\/strong><\/dt>\n<dd id=\"fs-id1165135190252\">a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135487256\" class=\"definition\">\n<dt><strong>piecewise function<\/strong><\/dt>\n<dd id=\"fs-id1165137452169\">a function in which more than one formula is used to define the output<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137863188\" class=\"definition\">\n<dt><strong>set-builder notation<\/strong><\/dt>\n<dd id=\"fs-id1165137863193\">a method of describing a set by a rule that all of its members obey; it takes the form [latex]\\left\\{x|\\text{statement about }x\\right\\}[\/latex]<\/dd>\n<\/dl>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5269\/2020\/06\/22204550\/stop-sign-with-hand-300x300.png\" alt=\"Stop Here\" width=\"300\" height=\"300\" \/><\/p>\n<h3 style=\"text-align: center;\"><span data-sheets-root=\"1\">STOP HERE and complete Homework 1.4 &#8211; Domain and Range<\/span><\/h3>\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-1792\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 60533, 61836, 47487, 11812. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 108347. <strong>Authored by<\/strong>: Coulston, Charles R. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 3190, 3191. <strong>Authored by<\/strong>: Anderson, Tophe. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 72181. <strong>Authored by<\/strong>: Carmichael, Patrick. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 30831, 32883. <strong>Authored by<\/strong>: Smart, Jim. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 92940, 3765, 93008. <strong>Authored by<\/strong>: Jenck, Michael. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 47481, 47483, 47484. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1657. <strong>Authored by<\/strong>: WebWork-Rochester, mb Sousa,James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Finding Function Values. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/NTmgEF_nZSc\">https:\/\/youtu.be\/NTmgEF_nZSc<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Ex: Evaluate a Function and Solve for a Function Value Given a Table. <strong>Authored by<\/strong>: Mathispower4u. <strong>Provided by<\/strong>: Phoenix College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/meqZdQkoNOQ\">https:\/\/youtu.be\/meqZdQkoNOQ<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Ex1: Evaluate a Function and Solve for a Function Value Given a Graph. <strong>Authored by<\/strong>: Mathispower4u. <strong>Provided by<\/strong>: Phoenix College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=_fO9gx1ncyg.\">https:\/\/www.youtube.com\/watch?v=_fO9gx1ncyg.<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Find Function Inputs for a Given Quadratic Function Output. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/GLOmTED1UwA\">https:\/\/youtu.be\/GLOmTED1UwA<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Write a Linear Relation as a Function. <strong>Authored by<\/strong>: James Sousa. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=lHTLjfPpFyQ&#038;feature=youtu.be\">https:\/\/www.youtube.com\/watch?v=lHTLjfPpFyQ&#038;feature=youtu.be<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and 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