{"id":82,"date":"2016-06-01T20:50:17","date_gmt":"2016-06-01T20:50:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=82"},"modified":"2016-10-03T21:05:54","modified_gmt":"2016-10-03T21:05:54","slug":"17-2-1-evaluating-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/17-2-1-evaluating-functions\/","title":{"raw":"Domain and Range","rendered":"Domain and Range"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Find the domain of a square root and rational function<\/li>\r\n \t<li>Find the domain and range of a function from the algebraic form.<\/li>\r\n \t<li>Define the domain of linear, quadratic, radical, and rational functions from graphs<\/li>\r\n<\/ul>\r\n<\/div>\r\nFunctions are a correspondence between two sets, called the <strong>domain<\/strong> and the <strong>range<\/strong>. When defining a function, you usually state what kind of numbers the domain (<i>x<\/i>) and range (<i>f(x)<\/i>) values can be. But even if you say they are real numbers, that doesn\u2019t mean that <i>all<\/i> real numbers can be used for <i>x<\/i>. It also doesn\u2019t mean that all real numbers can be function values, <i>f<\/i>(<i>x<\/i>). There may be restrictions on the domain and range. The restrictions partly depend on the <i>type<\/i> of function.\r\n\r\nIn this topic, all functions will be restricted to real number values. That is, only real numbers can be used in the domain, and only real numbers can be in the range.\r\n\r\nThere are two main reasons why domains are restricted.\r\n<ul>\r\n \t<li>You can\u2019t divide by 0.<\/li>\r\n \t<li>You can\u2019t 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\nIn what kind of functions would these two issues occur?\r\n<div>\r\n<ul>\r\n \t<li>the function is a rational function and the denominator is 0 for some value or values of <i>x,\u00a0<\/i>[latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex] is a rational function<\/li>\r\n \t<li>the function is a radical function with an even index (such as a square root), and the radicand can be negative for some value or values of <i>x<\/i>.\u00a0[latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex] is a radical function<\/li>\r\n<\/ul>\r\nThe following table gives examples of domain restrictions for several different rational functions.\r\n<table cellspacing=\"0\" cellpadding=\"0\">\r\n<thead>\r\n<tr>\r\n<th>Function<\/th>\r\n<th>Notes<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] f(x)=\\frac{1}{x}[\/latex]<\/td>\r\n<td>If [latex]x=0[\/latex], you would be dividing by 0, so [latex]x\\neq0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\frac{2+x}{x-3}[\/latex]<\/td>\r\n<td>If [latex]x=3[\/latex], you would be dividing by 0, so [latex]x\\neq3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\frac{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 0. So [latex]x\\neq1[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\frac{x+1}{{{x}^{2}}-1}[\/latex]<\/td>\r\n<td>Both [latex]x=1[\/latex] and [latex]x=\u22121[\/latex] would make the denominator 0. Again, this function can be simplified to [latex] f(x)=\\frac{1}{x-1}[\/latex], but when [latex]x=1[\/latex] or [latex]x=\u22121[\/latex] the <i>original<\/i> function would include division by 0, so [latex]x\\neq1[\/latex] and [latex]x\\neq\u22121[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x)=\\frac{2(x-1)}{{{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 0. The least it can be is 1, so there is no danger of division by 0.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<i>Square roots of negative numbers<\/i> could happen whenever the function has a variable under a radical with an even root. Look at these 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.\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], you would be taking the square root of a negative number, so [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], you 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 x is positive. (For example, if [latex]x=\u22123[\/latex], then [latex]\u2212x=3[\/latex]. If [latex]x=1[\/latex], then [latex]\u2212x=\u22121[\/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>\n\n[latex]x^{2}\u20131[\/latex] must be positive, [latex]x^{2}\u20131&gt;0[\/latex].\r\n\r\nSo [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>\n\nThere are no domain restrictions, even though there is a variable under the radical. Since\r\n\r\n[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\n<\/div>\r\nSo how, exactly do you define the domain of a function anyway?\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function written in equation form, 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 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<\/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. 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\">\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] . If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\r\n \t<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].\r\n\r\n[reveal-answer q=\"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]\\begin{cases}2-x=0\\hfill \\\\ -x=-2\\hfill \\\\ x=2\\hfill \\end{cases}[\/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[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img 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=\"Line graph of x=!2.\" width=\"487\" height=\"164\" data-media-type=\"image\/jpg\" \/> <b>Figure 3<\/b>[\/caption]\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\">\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<\/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\nSet the radicand greater than or equal to zero and solve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{cases}7-x\\ge 0\\hfill \\\\ -x\\ge -7\\hfill \\\\ x\\le 7\\hfill \\end{cases}[\/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&amp;feature=youtu.be\r\n\r\nThere can be functions in which the domain and range do not intersect at all.\u00a0For 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.\r\n<h2>Find Domain and Range From a Graph<\/h2>\r\nFinding domain and range of different functions is often a matter of asking yourself, what values can this function <i>not<\/i>\u00a0have? Pictures make it easier to visualize what domain and range are, so we will show how to define the domain and range of functions given their graphs.\r\n\r\nWhat are the domain and range of the real-valued function [latex]f(x)=x+3[\/latex]?\r\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Line for f(x)=x+3\" width=\"322\" height=\"353\" \/>\r\n\r\nAny real number can be substituted for <i>x<\/i> and get a meaningful output. For <i>any<\/i> real number, you can always find an <i>x<\/i> value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range.\r\nThe domain and range are all real numbers.\r\n\r\nFor the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat are the domain and range of the real-valued function [latex]f(x)=\u22123x^{2}+6x+1[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/>\r\n\r\n[reveal-answer q=\"223692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223692\"]This is a <i>quadratic <\/i>function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing that will restrict the domain. Any real number can be used for <i>x<\/i> to get a meaningful output.\r\n\r\nBecause the coefficient of [latex]x^{2}[\/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value.\r\n\r\nThe vertex, or high\u00a0point, is at (1, 4). From the graph, you can see that [latex]f(x)\\leq4[\/latex].\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers, and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\leq4[\/latex].\r\n\r\nYou can check that the vertex is indeed at (1, 4). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same <i>y<\/i> value. The vertex must lie on the line of reflection, because it\u2019s the only point that does not have a mirror image!\r\n\r\nIn the previous example, notice that when [latex]x=2[\/latex] and when [latex]x=0[\/latex], the function value is 1. (You can verify this by evaluating <i>f<\/i>(2) and <i>f<\/i>(0).) That is, both (2, 1) and (0, 1) are on the graph. The line of reflection here is [latex]x=1[\/latex], so the vertex must be at the point (1, <i>f<\/i>(1)). Evaluating <i>f<\/i>(1)<i> <\/i>gives [latex]f(1)=4[\/latex], so the vertex is at (1, 4).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat is the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Radical function stemming from negative 5, negative 2.\" width=\"308\" height=\"346\" \/>\r\n\r\n[reveal-answer q=\"231228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231228\"]This is a <i>radical <\/i>function. The domain of a radical function is any <i>x<\/i> value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\\geq0[\/latex], so [latex]x\\geq\u22125[\/latex].\r\n\r\nSince the square root must always be positive or 0, [latex] \\displaystyle \\sqrt{x+5}\\ge 0[\/latex]. That means [latex] \\displaystyle -2+\\sqrt{x+5}\\ge -2[\/latex].\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers <i>x<\/i> where [latex]x\\geq\u22125[\/latex], and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\geq\u22122[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat is the domain of the real-valued function [latex] \\displaystyle f(x)=\\frac{3x}{x+2}[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232531\/image049.gif\" alt=\"Rational function\" width=\"310\" height=\"321\" \/>\r\n\r\n[reveal-answer q=\"666335\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"666335\"]This is a <i>rational <\/i>function. The domain of a rational function is restricted where the denominator is 0. In this case, [latex]x+2[\/latex] is the denominator, and this is 0 only when [latex]x=\u22122[\/latex].\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers except [latex]\u22122[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show how to define the domain and range of\u00a0functions from their graphs.\r\n\r\nhttps:\/\/youtu.be\/QAxZEelInJc\r\n<h2>Summary<\/h2>\r\n<i>Division by 0<\/i> could happen whenever the function has a variable in the <i>denominator <\/i>of a rational expression. That is, it\u2019s something to look for in <i>rational functions.<\/i> Look at these examples, and note that \u201cdivision by 0\u201d doesn\u2019t necessarily mean that <i>x<\/i> is 0!\r\n\r\nAlthough a function may be given as \u201creal valued,\u201d it may be that the function has restrictions to its domain and range. There may be some real numbers that can\u2019t be part of the domain or part of the range. This is particularly true with rational and radical functions, which can have restrictions to domain, range, or both. Other functions, such as quadratic functions and polynomial functions of even degree, also can have restrictions to their range.\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Find the domain of a square root and rational function<\/li>\n<li>Find the domain and range of a function from the algebraic form.<\/li>\n<li>Define the domain of linear, quadratic, radical, and rational functions from graphs<\/li>\n<\/ul>\n<\/div>\n<p>Functions are a correspondence between two sets, called the <strong>domain<\/strong> and the <strong>range<\/strong>. When defining a function, you usually state what kind of numbers the domain (<i>x<\/i>) and range (<i>f(x)<\/i>) values can be. But even if you say they are real numbers, that doesn\u2019t mean that <i>all<\/i> real numbers can be used for <i>x<\/i>. It also doesn\u2019t mean that all real numbers can be function values, <i>f<\/i>(<i>x<\/i>). There may be restrictions on the domain and range. The restrictions partly depend on the <i>type<\/i> of function.<\/p>\n<p>In this topic, all functions will be restricted to real number values. That is, only real numbers can be used in the domain, and only real numbers can be in the range.<\/p>\n<p>There are two main reasons why domains are restricted.<\/p>\n<ul>\n<li>You can\u2019t divide by 0.<\/li>\n<li>You can\u2019t 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>In what kind of functions would these two issues occur?<\/p>\n<div>\n<ul>\n<li>the function is a rational function and the denominator is 0 for some value or values of <i>x,\u00a0<\/i>[latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex] is a rational function<\/li>\n<li>the function is a radical function with an even index (such as a square root), and the radicand can be negative for some value or values of <i>x<\/i>.\u00a0[latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex] is a radical function<\/li>\n<\/ul>\n<p>The following table gives examples of domain restrictions for several different rational functions.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<thead>\n<tr>\n<th>Function<\/th>\n<th>Notes<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f(x)=\\frac{1}{x}[\/latex]<\/td>\n<td>If [latex]x=0[\/latex], you would be dividing by 0, so [latex]x\\neq0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\frac{2+x}{x-3}[\/latex]<\/td>\n<td>If [latex]x=3[\/latex], you would be dividing by 0, so [latex]x\\neq3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\frac{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 0. So [latex]x\\neq1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\frac{x+1}{{{x}^{2}}-1}[\/latex]<\/td>\n<td>Both [latex]x=1[\/latex] and [latex]x=\u22121[\/latex] would make the denominator 0. Again, this function can be simplified to [latex]f(x)=\\frac{1}{x-1}[\/latex], but when [latex]x=1[\/latex] or [latex]x=\u22121[\/latex] the <i>original<\/i> function would include division by 0, so [latex]x\\neq1[\/latex] and [latex]x\\neq\u22121[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)=\\frac{2(x-1)}{{{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 0. The least it can be is 1, so there is no danger of division by 0.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><i>Square roots of negative numbers<\/i> could happen whenever the function has a variable under a radical with an even root. Look at these 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], you would be taking the square root of a negative number, so [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], you 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 x is positive. (For example, if [latex]x=\u22123[\/latex], then [latex]\u2212x=3[\/latex]. If [latex]x=1[\/latex], then [latex]\u2212x=\u22121[\/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>\n<p>[latex]x^{2}\u20131[\/latex] must be positive, [latex]x^{2}\u20131>0[\/latex].<\/p>\n<p>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>\n<p>There are no domain restrictions, even though there is a variable under the radical. Since<\/p>\n<p>[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<\/div>\n<p>So how, exactly do you define the domain of a function anyway?<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function written in equation form, find the domain.<\/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<\/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. 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>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] . If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\n<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"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]\\begin{cases}2-x=0\\hfill \\\\ -x=-2\\hfill \\\\ x=2\\hfill \\end{cases}[\/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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200611\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\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-1\" title=\"Ex:  The Domain of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"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<\/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>Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{cases}7-x\\ge 0\\hfill \\\\ -x\\ge -7\\hfill \\\\ x\\le 7\\hfill \\end{cases}[\/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-2\" title=\"Ex: Domain and Range of Square Root Functions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>There can be functions in which the domain and range do not intersect at all.\u00a0For 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.<\/p>\n<h2>Find Domain and Range From a Graph<\/h2>\n<p>Finding domain and range of different functions is often a matter of asking yourself, what values can this function <i>not<\/i>\u00a0have? Pictures make it easier to visualize what domain and range are, so we will show how to define the domain and range of functions given their graphs.<\/p>\n<p>What are the domain and range of the real-valued function [latex]f(x)=x+3[\/latex]?<br \/>\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Line for f(x)=x+3\" width=\"322\" height=\"353\" \/><\/p>\n<p>Any real number can be substituted for <i>x<\/i> and get a meaningful output. For <i>any<\/i> real number, you can always find an <i>x<\/i> value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range.<br \/>\nThe domain and range are all real numbers.<\/p>\n<p>For the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What are the domain and range of the real-valued function [latex]f(x)=\u22123x^{2}+6x+1[\/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\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223692\">Show Solution<\/span><\/p>\n<div id=\"q223692\" class=\"hidden-answer\" style=\"display: none\">This is a <i>quadratic <\/i>function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing that will restrict the domain. Any real number can be used for <i>x<\/i> to get a meaningful output.<\/p>\n<p>Because the coefficient of [latex]x^{2}[\/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value.<\/p>\n<p>The vertex, or high\u00a0point, is at (1, 4). From the graph, you can see that [latex]f(x)\\leq4[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers, and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\leq4[\/latex].<\/p>\n<p>You can check that the vertex is indeed at (1, 4). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same <i>y<\/i> value. The vertex must lie on the line of reflection, because it\u2019s the only point that does not have a mirror image!<\/p>\n<p>In the previous example, notice that when [latex]x=2[\/latex] and when [latex]x=0[\/latex], the function value is 1. (You can verify this by evaluating <i>f<\/i>(2) and <i>f<\/i>(0).) That is, both (2, 1) and (0, 1) are on the graph. The line of reflection here is [latex]x=1[\/latex], so the vertex must be at the point (1, <i>f<\/i>(1)). Evaluating <i>f<\/i>(1)<i> <\/i>gives [latex]f(1)=4[\/latex], so the vertex is at (1, 4).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What is the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}[\/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\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Radical function stemming from negative 5, negative 2.\" width=\"308\" height=\"346\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q231228\">Show Solution<\/span><\/p>\n<div id=\"q231228\" class=\"hidden-answer\" style=\"display: none\">This is a <i>radical <\/i>function. The domain of a radical function is any <i>x<\/i> value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\\geq0[\/latex], so [latex]x\\geq\u22125[\/latex].<\/p>\n<p>Since the square root must always be positive or 0, [latex]\\displaystyle \\sqrt{x+5}\\ge 0[\/latex]. That means [latex]\\displaystyle -2+\\sqrt{x+5}\\ge -2[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers <i>x<\/i> where [latex]x\\geq\u22125[\/latex], and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\geq\u22122[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What is the domain of the real-valued function [latex]\\displaystyle f(x)=\\frac{3x}{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\/121\/2016\/04\/18232531\/image049.gif\" alt=\"Rational function\" width=\"310\" height=\"321\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q666335\">Show Solution<\/span><\/p>\n<div id=\"q666335\" class=\"hidden-answer\" style=\"display: none\">This is a <i>rational <\/i>function. The domain of a rational function is restricted where the denominator is 0. In this case, [latex]x+2[\/latex] is the denominator, and this is 0 only when [latex]x=\u22122[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers except [latex]\u22122[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show how to define the domain and range of\u00a0functions from their graphs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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<h2>Summary<\/h2>\n<p><i>Division by 0<\/i> could happen whenever the function has a variable in the <i>denominator <\/i>of a rational expression. That is, it\u2019s something to look for in <i>rational functions.<\/i> Look at these examples, and note that \u201cdivision by 0\u201d doesn\u2019t necessarily mean that <i>x<\/i> is 0!<\/p>\n<p>Although a function may be given as \u201creal valued,\u201d it may be that the function has restrictions to its domain and range. There may be some real numbers that can\u2019t be part of the domain or part of the range. This is particularly true with rational and radical functions, which can have restrictions to domain, range, or both. Other functions, such as quadratic functions and polynomial functions of even degree, also can have restrictions to their range.<\/p>\n<h2><\/h2>\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-82\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <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\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for Free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/li><li>Unit 17: Functions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Determine the Domain and Range of the Graph of a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QAxZEelInJc\">https:\/\/youtu.be\/QAxZEelInJc<\/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":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for Free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\"},{\"type\":\"cc\",\"description\":\"Unit 17: Functions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute 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