{"id":8015,"date":"2021-01-13T04:37:45","date_gmt":"2021-01-13T04:37:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&#038;p=8015"},"modified":"2026-02-11T00:43:32","modified_gmt":"2026-02-11T00:43:32","slug":"8-1-revisiting-domain","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/8-1-revisiting-domain\/","title":{"raw":"8.1: Revisiting Domain","rendered":"8.1: Revisiting Domain"},"content":{"raw":"**As of Fall 2023, Math 0990 is no longer using this module. Students working on an \"Incomplete\" prior to Fall 2023 will still need this module**\r\n<div class=\"textbox learning-objectives\">\r\n<h3>section 8.1 Learning Objectives<\/h3>\r\n<strong>8.1: Revisiting Domain<\/strong>\r\n<ul>\r\n \t<li>Determine the domain of a function given an equation\r\n<ul>\r\n \t<li>Find the domain of a polynomial<\/li>\r\n \t<li>Find the domain of a square root function<\/li>\r\n \t<li>Find the domain of a non-even root<\/li>\r\n \t<li>Find the domain of a rational function<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nRecall from section <a href=\"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-1-intro-functions\/\">3.2<\/a>, <strong>functions<\/strong> are a correspondence between two sets, called the <strong>domain<\/strong> and the <strong>range<\/strong>. We previously practiced finding the domain and range of a set of ordered pairs, but now we will look at finding the domain when given the equation of a function.\r\n\r\nWhen defining a function, we usually state what kind of numbers the domain and range values can be. But even if we say the values are real numbers, that does not mean that <strong><i>all<\/i><\/strong>\u00a0real numbers can be used for [latex]x[\/latex].\u00a0It also does not mean that all real numbers can be function values, [latex]f(x)[\/latex]. There may be restrictions on the domain (and range). The restrictions partly depend on the <i>type<\/i> of function.\r\n<h2>Restrictions on Domain<\/h2>\r\nThere are two main reasons domain will be restricted for a function (that we focus on in this course). Any input value that would ask the function to violate one (or both) of the rules below, creates a restriction on the domain because those values cannot be allowed to be input into the function.\u00a0 (Hint: If you try to <em>input\u00a0<\/em>either of these things in your calculator, it will give you a \"Domain Error\" message as the <em>output<\/em> message).\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><strong>Reasons for Restricting the Domain of a function<\/strong><\/h3>\r\n<ul>\r\n \t<li>\r\n<h2>We cannot take the square (or other even) root of a negative number. The result will not be a real number.<\/h2>\r\n<\/li>\r\n \t<li>\r\n<h2>We cannot divide by\u00a0[latex]0[\/latex].<\/h2>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nThese two issues would be of concern when the function is either of the following:\r\n<div>\r\n<ul>\r\n \t<li style=\"margin-top: 0.5em;\">A\u00a0<strong>radical function<\/strong>\u00a0with an even index (such as a square root), where the radicand (quantity under the radical) could potentially be negative for some value or values of\u00a0<i>x<\/i>.\u00a0[latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex] is a radical function.<\/li>\r\n \t<li>A <strong>rational function<\/strong> where the denominator could potentially become [latex]0[\/latex] for some value or values of <i>x,\u00a0<\/i>[latex]f\\left(x\\right)=\\dfrac{5}{2-x}[\/latex] is an example of a rational function.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: start; font-size: 16px;\">The following table gives examples of domain restrictions for several different <strong>radical functions<\/strong>.<i>\u00a0Roots of negative numbers<\/i>\u00a0will be an issue whenever the function has a variable under a radical with an even root. Look at the following examples and note that \u201csquare root of a negative variable\u201d doesn\u2019t necessarily mean that the value under the radical sign is negative. For example, if [latex]x=\u22124[\/latex], then [latex]\u2212x=\u2212(\u22124)=4[\/latex], a positive number.<\/p>\r\n\r\n<table cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr>\r\n<th>Function<\/th>\r\n<th>Restrictions to the Domain<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] f(x)=\\sqrt{x}[\/latex]<\/td>\r\n<td>If [latex]x&lt;0[\/latex], we would be taking the square root of a negative number, so instead of allowing [latex]x[\/latex] to be any real number, we restrict the domain down to [latex]x\\geq0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\sqrt{x+10}[\/latex]<\/td>\r\n<td>If [latex]x&lt;\u221210[\/latex], we would be taking the square root of a negative number, so [latex]x\\geq\u221210[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\sqrt{-x}[\/latex]<\/td>\r\n<td>When is [latex]-x[\/latex] negative? Only when [latex]x[\/latex] is positive. (For example, if [latex]x=1[\/latex], then [latex]\u2212x=-1[\/latex]. But if [latex]x=-3[\/latex], then [latex]\u2212x=-(-3)=3[\/latex].)This means [latex]x\\leq0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\sqrt{{{x}^{2}}-1}[\/latex]<\/td>\r\n<td>[latex]x^{2}\u20131[\/latex] must be positive, [latex]x^{2}\u20131&gt;0[\/latex]. So [latex]x^{2}&gt;1[\/latex]. This happens only when x is greater than 1 or less than [latex]\u22121[\/latex]:\u00a0[latex]x\\leq\u22121[\/latex] or [latex]x\\geq1[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\sqrt{{{x}^{2}}+10}[\/latex]<\/td>\r\n<td>There are no domain restrictions even though there is a variable under the radical. Since [latex]x^{2}\\ge0[\/latex], [latex]x^{2}+10[\/latex]\u00a0can never be negative. The least it can be is [latex]\\sqrt{10}[\/latex], so there is no danger of taking the square root of a negative number.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following table gives examples of domain restrictions for several different <strong>rational functions<\/strong>. Note that a rational function has the variable present in the denominator.\r\n<table cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr>\r\n<th>Function<\/th>\r\n<th>Restrictions to the Domain<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] f(x)=\\dfrac{1}{x}[\/latex]<\/td>\r\n<td>If [latex]x=0[\/latex], we would be dividing by\u00a0[latex]0[\/latex], so [latex]x\\neq0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\dfrac{2+x}{x-3}[\/latex]<\/td>\r\n<td>If [latex]x=3[\/latex], we would be dividing by\u00a0[latex]0[\/latex], so [latex]x\\neq3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\dfrac{2(x-1)}{x-1}[\/latex]<\/td>\r\n<td>Although you can simplify this function to [latex]f(x)=2[\/latex], when [latex]x=1[\/latex], the original function would include division by\u00a0[latex]0[\/latex], so [latex]x\\neq1[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\dfrac{x}{{{x}^{2}}-1}[\/latex]<\/td>\r\n<td>Both [latex]x=1[\/latex] and [latex]x=\u22121[\/latex] would make the denominator 0,\u00a0so [latex]x\\neq1[\/latex] and [latex]x\\neq\u22121[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\dfrac{2}{{{x}^{2}}+1}[\/latex]<\/td>\r\n<td>This is an example with <i>no <\/i>domain<i> <\/i>restrictions, even though there is a variable in the denominator. Since\u00a0[latex]x^{2}\\geq0,x^{2}+1[\/latex] can never be\u00a0[latex]0[\/latex]. The least it can be is\u00a0[latex]1[\/latex], so there is no danger of division by\u00a0[latex]0[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"font-size: 1rem; text-align: initial;\">So, how exactly do you define the domain of a function?<\/span>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">How To find the domain,\u00a0<strong>Given a function written in equation form<\/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.\r\n<ul>\r\n \t<li>Is there an even index on a radical in the function? <em>If so there may be restrictions on values related to this.<\/em><\/li>\r\n \t<li>Is there a fraction with a variable in the denominator in the function?\u00a0<em>If so there may be restrictions on values related to this.<\/em><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Write the domain in interval notation (or Set-Builder Notation), if possible.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 1<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].\r\n\r\n[reveal-answer q=\"480036\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"480036\"]\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. Notice, no matter what the input is, it will never\u00a0violate either of the rules,\r\n<ul>\r\n \t<li>We cannot divide by\u00a0[latex]0[\/latex].<\/li>\r\n \t<li>We cannot take the square (or other even) root of a negative number, as the result will not be a real number.<\/li>\r\n<\/ul>\r\nThe 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\">\r\n<h3 style=\"text-align: center;\">How To find the domain, Given a function written in equation form including an <strong>even<\/strong> root<\/h3>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Since there is an even root, we want to\u00a0exclude any real numbers that result in a negative number in the radicand from being in the domain. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex]. (Any numbers <em>not<\/em> in the solution are values restricted from the domain.)<\/li>\r\n \t<li>The solution(s) from step 2 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 2<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].\r\n\r\n[reveal-answer q=\"275485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"275485\"]\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\nIn other words, we set the radicand greater than or equal to zero and solve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]7-x\\ge 0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-x\\ge -7[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x\\le 7[\/latex]<\/p>\r\nThe answer above is our domain as it excludes 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\n<div class=\"textbox exercises\">\r\n<h3>Example 3<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt[3]{-2x-3}[\/latex].\r\n\r\n[reveal-answer q=\"721404\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"721404\"]\r\n\r\nWe know from the previous examples that there are restrictions on what values the variable can be when the root is an even root.\u00a0 With even roots, the radicand cannot be a negative number.\u00a0 However, with odd roots, we don\u2019t have the same restriction.\u00a0 The radicand with odd roots can be positive, negative, or [latex]0[\/latex].\u00a0 Therefore, the domain of this function is the set of all real numbers.\u00a0 In interval notation this would be represented by [latex](-\\infty,\\infty)[\/latex].\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nBelow is another example of finding the domain of a radical function:\r\n\r\n[embed]https:\/\/youtu.be\/4h54s7BBPpA[\/embed]\r\n<div class=\"textbox\">\r\n<h3 style=\"text-align: center;\">How To find the domain, Given a function written in equation form that includes a fraction<\/h3>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input.\r\n<ul>\r\n \t<li>If there is a denominator in the function\u2019s formula with variables in it, set the denominator equal to zero and solve for [latex]x[\/latex] . These values are restricted from the domain.<\/li>\r\n \t<li>If the function also contains an <strong>even<\/strong> root, set the radicand greater than or equal to\u00a0[latex]0[\/latex] and then solve. Any numbers not in the solution are restricted values from the domain.<\/li>\r\n<\/ul>\r\n<\/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 4<\/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=\"995188\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"995188\"]\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]2-x=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-x=-2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=2[\/latex]<\/p>\r\nNow, we will exclude\u00a0[latex]2[\/latex] 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-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200611\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Number line. Open circle at 2 and highlighted to left and right of 2. Expression: x &lt; 2 or x &gt; 2. Line 2 Expression: (negative infinity, 2) union (2, infinity).\" 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\nAt times you may be asked to express your solution in Set-Builder Notation. Recall in Section 2.1, we were introduced to Set-Builder Notation.\u00a0 We can use that notation here to describe the domain of the above function.\u00a0 The domain of this function written in Set-Builder Notation would be:\r\n<p style=\"text-align: center;\">[latex]\\{x|x\\neq2\\}[\/latex]<\/p>\r\nThis would be read as \u201cThe set of all [latex]x[\/latex] such that [latex]x\\neq2[\/latex] .\u201d\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the video below for another description of domain as well as some more examples involving radical functions and rational functions:\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=F0HWrBHP58k&amp;feature=emb_title[\/embed]","rendered":"<p>**As of Fall 2023, Math 0990 is no longer using this module. Students working on an &#8220;Incomplete&#8221; prior to Fall 2023 will still need this module**<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>section 8.1 Learning Objectives<\/h3>\n<p><strong>8.1: Revisiting Domain<\/strong><\/p>\n<ul>\n<li>Determine the domain of a function given an equation\n<ul>\n<li>Find the domain of a polynomial<\/li>\n<li>Find the domain of a square root function<\/li>\n<li>Find the domain of a non-even root<\/li>\n<li>Find the domain of a rational function<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Recall from section <a href=\"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-1-intro-functions\/\">3.2<\/a>, <strong>functions<\/strong> are a correspondence between two sets, called the <strong>domain<\/strong> and the <strong>range<\/strong>. We previously practiced finding the domain and range of a set of ordered pairs, but now we will look at finding the domain when given the equation of a function.<\/p>\n<p>When defining a function, we usually state what kind of numbers the domain and range values can be. But even if we say the values are real numbers, that does not mean that <strong><i>all<\/i><\/strong>\u00a0real numbers can be used for [latex]x[\/latex].\u00a0It also does not mean that all real numbers can be function values, [latex]f(x)[\/latex]. There may be restrictions on the domain (and range). The restrictions partly depend on the <i>type<\/i> of function.<\/p>\n<h2>Restrictions on Domain<\/h2>\n<p>There are two main reasons domain will be restricted for a function (that we focus on in this course). Any input value that would ask the function to violate one (or both) of the rules below, creates a restriction on the domain because those values cannot be allowed to be input into the function.\u00a0 (Hint: If you try to <em>input\u00a0<\/em>either of these things in your calculator, it will give you a &#8220;Domain Error&#8221; message as the <em>output<\/em> message).<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><strong>Reasons for Restricting the Domain of a function<\/strong><\/h3>\n<ul>\n<li>\n<h2>We cannot take the square (or other even) root of a negative number. The result will not be a real number.<\/h2>\n<\/li>\n<li>\n<h2>We cannot divide by\u00a0[latex]0[\/latex].<\/h2>\n<\/li>\n<\/ul>\n<\/div>\n<p>These two issues would be of concern when the function is either of the following:<\/p>\n<div>\n<ul>\n<li style=\"margin-top: 0.5em;\">A\u00a0<strong>radical function<\/strong>\u00a0with an even index (such as a square root), where the radicand (quantity under the radical) could potentially be negative for some value or values of\u00a0<i>x<\/i>.\u00a0[latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex] is a radical function.<\/li>\n<li>A <strong>rational function<\/strong> where the denominator could potentially become [latex]0[\/latex] for some value or values of <i>x,\u00a0<\/i>[latex]f\\left(x\\right)=\\dfrac{5}{2-x}[\/latex] is an example of a rational function.<\/li>\n<\/ul>\n<p style=\"text-align: start; font-size: 16px;\">The following table gives examples of domain restrictions for several different <strong>radical functions<\/strong>.<i>\u00a0Roots of negative numbers<\/i>\u00a0will be an issue whenever the function has a variable under a radical with an even root. Look at the following examples and note that \u201csquare root of a negative variable\u201d doesn\u2019t necessarily mean that the value under the radical sign is negative. For example, if [latex]x=\u22124[\/latex], then [latex]\u2212x=\u2212(\u22124)=4[\/latex], a positive number.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr>\n<th>Function<\/th>\n<th>Restrictions to the Domain<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f(x)=\\sqrt{x}[\/latex]<\/td>\n<td>If [latex]x<0[\/latex], we would be taking the square root of a negative number, so instead of allowing [latex]x[\/latex] to be any real number, we restrict the domain down to [latex]x\\geq0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\sqrt{x+10}[\/latex]<\/td>\n<td>If [latex]x<\u221210[\/latex], we would be taking the square root of a negative number, so [latex]x\\geq\u221210[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\sqrt{-x}[\/latex]<\/td>\n<td>When is [latex]-x[\/latex] negative? Only when [latex]x[\/latex] is positive. (For example, if [latex]x=1[\/latex], then [latex]\u2212x=-1[\/latex]. But if [latex]x=-3[\/latex], then [latex]\u2212x=-(-3)=3[\/latex].)This means [latex]x\\leq0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\sqrt{{{x}^{2}}-1}[\/latex]<\/td>\n<td>[latex]x^{2}\u20131[\/latex] must be positive, [latex]x^{2}\u20131>0[\/latex]. So [latex]x^{2}>1[\/latex]. This happens only when x is greater than 1 or less than [latex]\u22121[\/latex]:\u00a0[latex]x\\leq\u22121[\/latex] or [latex]x\\geq1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\sqrt{{{x}^{2}}+10}[\/latex]<\/td>\n<td>There are no domain restrictions even though there is a variable under the radical. Since [latex]x^{2}\\ge0[\/latex], [latex]x^{2}+10[\/latex]\u00a0can never be negative. The least it can be is [latex]\\sqrt{10}[\/latex], so there is no danger of taking the square root of a negative number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following table gives examples of domain restrictions for several different <strong>rational functions<\/strong>. Note that a rational function has the variable present in the denominator.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr>\n<th>Function<\/th>\n<th>Restrictions to the Domain<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f(x)=\\dfrac{1}{x}[\/latex]<\/td>\n<td>If [latex]x=0[\/latex], we would be dividing by\u00a0[latex]0[\/latex], so [latex]x\\neq0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\dfrac{2+x}{x-3}[\/latex]<\/td>\n<td>If [latex]x=3[\/latex], we would be dividing by\u00a0[latex]0[\/latex], so [latex]x\\neq3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\dfrac{2(x-1)}{x-1}[\/latex]<\/td>\n<td>Although you can simplify this function to [latex]f(x)=2[\/latex], when [latex]x=1[\/latex], the original function would include division by\u00a0[latex]0[\/latex], so [latex]x\\neq1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\dfrac{x}{{{x}^{2}}-1}[\/latex]<\/td>\n<td>Both [latex]x=1[\/latex] and [latex]x=\u22121[\/latex] would make the denominator 0,\u00a0so [latex]x\\neq1[\/latex] and [latex]x\\neq\u22121[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\dfrac{2}{{{x}^{2}}+1}[\/latex]<\/td>\n<td>This is an example with <i>no <\/i>domain<i> <\/i>restrictions, even though there is a variable in the denominator. Since\u00a0[latex]x^{2}\\geq0,x^{2}+1[\/latex] can never be\u00a0[latex]0[\/latex]. The least it can be is\u00a0[latex]1[\/latex], so there is no danger of division by\u00a0[latex]0[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"font-size: 1rem; text-align: initial;\">So, how exactly do you define the domain of a function?<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">How To find the domain,\u00a0<strong>Given a function written in equation form<\/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.\n<ul>\n<li>Is there an even index on a radical in the function? <em>If so there may be restrictions on values related to this.<\/em><\/li>\n<li>Is there a fraction with a variable in the denominator in the function?\u00a0<em>If so there may be restrictions on values related to this.<\/em><\/li>\n<\/ul>\n<\/li>\n<li>Write the domain in interval notation (or Set-Builder Notation), if possible.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 1<\/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=\"q480036\">Show Solution<\/span><\/p>\n<div id=\"q480036\" 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. Notice, no matter what the input is, it will never\u00a0violate either of the rules,<\/p>\n<ul>\n<li>We cannot divide by\u00a0[latex]0[\/latex].<\/li>\n<li>We cannot take the square (or other even) root of a negative number, as the result will not be a real number.<\/li>\n<\/ul>\n<p>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\">\n<h3 style=\"text-align: center;\">How To find the domain, Given a function written in equation form including an <strong>even<\/strong> root<\/h3>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Since there is an even root, we want to\u00a0exclude any real numbers that result in a negative number in the radicand from being in the domain. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex]. (Any numbers <em>not<\/em> in the solution are values restricted from the domain.)<\/li>\n<li>The solution(s) from step 2 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 2<\/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=\"q275485\">Show Solution<\/span><\/p>\n<div id=\"q275485\" 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>In other words, we set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]7-x\\ge 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-x\\ge -7[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x\\le 7[\/latex]<\/p>\n<p>The answer above is our domain as it excludes any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[\/latex], or [latex]\\left(-\\infty ,7\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 3<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt[3]{-2x-3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q721404\">Show Answer<\/span><\/p>\n<div id=\"q721404\" class=\"hidden-answer\" style=\"display: none\">\n<p>We know from the previous examples that there are restrictions on what values the variable can be when the root is an even root.\u00a0 With even roots, the radicand cannot be a negative number.\u00a0 However, with odd roots, we don\u2019t have the same restriction.\u00a0 The radicand with odd roots can be positive, negative, or [latex]0[\/latex].\u00a0 Therefore, the domain of this function is the set of all real numbers.\u00a0 In interval notation this would be represented by [latex](-\\infty,\\infty)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Below is another example of finding the domain of a radical function:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Domain of a radical function | Functions and their graphs | Algebra II | Khan Academy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/4h54s7BBPpA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox\">\n<h3 style=\"text-align: center;\">How To find the domain, Given a function written in equation form that includes a fraction<\/h3>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input.\n<ul>\n<li>If there is a denominator in the function\u2019s formula with variables in it, set the denominator equal to zero and solve for [latex]x[\/latex] . These values are restricted from the domain.<\/li>\n<li>If the function also contains an <strong>even<\/strong> root, set the radicand greater than or equal to\u00a0[latex]0[\/latex] and then solve. Any numbers not in the solution are restricted values from the domain.<\/li>\n<\/ul>\n<\/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 4<\/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=\"q995188\">Show Solution<\/span><\/p>\n<div id=\"q995188\" 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]2-x=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-x=-2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=2[\/latex]<\/p>\n<p>Now, we will exclude\u00a0[latex]2[\/latex] 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-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200611\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Number line. Open circle at 2 and highlighted to left and right of 2. Expression: x &lt; 2 or x &gt; 2. Line 2 Expression: (negative infinity, 2) union (2, infinity).\" 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<p>At times you may be asked to express your solution in Set-Builder Notation. Recall in Section 2.1, we were introduced to Set-Builder Notation.\u00a0 We can use that notation here to describe the domain of the above function.\u00a0 The domain of this function written in Set-Builder Notation would be:<\/p>\n<p style=\"text-align: center;\">[latex]\\{x|x\\neq2\\}[\/latex]<\/p>\n<p>This would be read as \u201cThe set of all [latex]x[\/latex] such that [latex]x\\neq2[\/latex] .\u201d<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the video below for another description of domain as well as some more examples involving radical functions and rational functions:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Finding the Domain of Rational and Radical Functions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/F0HWrBHP58k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-8015\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Jay Abrams et, al.. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li><strong>Authored by<\/strong>: mathteachernw. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=F0HWrBHP58k&#038;feature=emb_title\">https:\/\/www.youtube.com\/watch?v=F0HWrBHP58k&#038;feature=emb_title<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">Public domain content<\/div><ul class=\"citation-list\"><li><strong>Authored by<\/strong>: Salman Khan. <strong>Provided by<\/strong>: Khan Academy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/4h54s7BBPpA\">https:\/\/youtu.be\/4h54s7BBPpA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":348856,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abrams et, 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