{"id":10836,"date":"2015-07-10T22:33:11","date_gmt":"2015-07-10T22:33:11","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&#038;p=10836"},"modified":"2018-07-07T20:13:46","modified_gmt":"2018-07-07T20:13:46","slug":"find-the-domain-of-a-function-defined-by-an-equation-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/find-the-domain-of-a-function-defined-by-an-equation-2\/","title":{"raw":"Find the domain of a function defined by an equation","rendered":"Find the domain of a function defined by an equation"},"content":{"raw":"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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_0022.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/> <b>Figure 2<\/b>[\/caption]\r\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \"holding area\" that contains \"raw materials\" for a \"function machine\" and the range as another \"holding area\" for the machine\u2019s products.<span id=\"fs-id1165137737552\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137761714\">We can write the <strong>domain and range<\/strong> in <strong>interval notation<\/strong>, which uses values within brackets 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>\r\n<p id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\r\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\r\n\r\n<ul id=\"fs-id1165135673417\">\r\n \t<li>The 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<p id=\"fs-id1165137807384\">The table below gives\u00a0a summary of interval notation.<span id=\"fs-id1165137406680\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/><\/span><\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\r\nFind the domain of the following function: [latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex] .\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Solution<\/h3>\r\nFirst identify the input values. The input value is the first coordinate in an <strong>ordered pair<\/strong>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.\r\n<div style=\"text-align: center\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 1<\/h3>\r\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\r\n<p id=\"fs-id1165137466017\" style=\"text-align: center\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\r\n<a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134225655\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165134355557\"><strong>How To: Given a function written in equation form, find the domain.<\/strong><\/h3>\r\n<ol id=\"fs-id1165134187286\">\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\r\n \t<li>Write the domain in interval form, if possible.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Finding the Domain of a Function<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135684349\" class=\"solution textbox shaded\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137594433\">The input value, shown by the variable [latex]x[\/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.<\/p>\r\n<p id=\"fs-id1165135309759\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 2<\/h3>\r\nFind the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}[\/latex].\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137417188\" class=\"note precalculus howto textbox\">\r\n<h3>How To:\u00a0Given a function written in an equation form that includes a fraction, find the domain.<\/h3>\r\n<ol id=\"fs-id1165137463251\">\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for [latex]x[\/latex] . If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\r\n \t<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Finding the Domain of a Function Involving a Denominator (Rational Function)<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135641743\" class=\"solution textbox shaded\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137565519\">When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137736620\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}2-x=0\\hfill \\\\ -x=-2\\hfill \\\\ x=2\\hfill \\end{cases}[\/latex]<\/div>\r\nNow, we will exclude 2 from the domain. The answers are all real numbers where [latex]x&lt;2[\/latex] or [latex]x&gt;2[\/latex]. We can use a symbol known as the union, [latex]\\cup [\/latex], to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<span>\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010544\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/> <b>Figure 3<\/b>[\/caption]\r\n<p id=\"fs-id1165134036054\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 3<\/h3>\r\nFind the domain of the function: [latex]f\\left(x\\right)=\\frac{1+4x}{2x - 1}[\/latex].\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135527005\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137733733\">How To: Given a function written in equation form including an even root, find the domain.<\/h3>\r\n<ol id=\"fs-id1165137820030\">\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/li>\r\n \t<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Finding the Domain of a Function with an Even Root<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137451129\" class=\"solution textbox shaded\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137453224\">When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.<\/p>\r\n<p id=\"fs-id1165137749755\">Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137727831\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}7-x\\ge 0\\hfill \\\\ -x\\ge -7\\hfill \\\\ x\\le 7\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137422794\">Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[\/latex], or [latex]\\left(-\\infty ,7\\right][\/latex].<\/p>\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 4<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<div class=\"\u201ctextbox\u201d textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can there be functions in which the domain and range do not intersect at all?<\/strong>\r\n<p id=\"fs-id1165137937737\"><em>Yes. For example, the function [latex]f\\left(x\\right)=-\\frac{1}{\\sqrt{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\r\n\r\n<\/div>","rendered":"<p>In Functions and Function Notation, we were introduced to the concepts of <strong>domain and range<\/strong>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_0022.jpg\" alt=\"Diagram of how a function relates two relations.\" width=\"487\" height=\"188\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135453892\">We can visualize the domain as a &#8220;holding area&#8221; that contains &#8220;raw materials&#8221; for a &#8220;function machine&#8221; and the range as another &#8220;holding area&#8221; for the machine\u2019s products.<span id=\"fs-id1165137737552\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137761714\">We can write the <strong>domain and range<\/strong> in <strong>interval notation<\/strong>, which uses values within brackets 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 id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\n<ul id=\"fs-id1165135673417\">\n<li>The 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 id=\"fs-id1165137807384\">The table below gives\u00a0a summary of interval notation.<span id=\"fs-id1165137406680\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010543\/CNX_Precalc_Figure_01_02_029n2.jpg\" alt=\"Summary of interval notation. Row 1, Inequality: x is greater than a. Interval notation: open parenthesis, a, infinity, close parenthesis. Row 2, Inequality: x is less than a. Interval notation: open parenthesis, negative infinity, a, close parenthesis. Row 3, Inequality x is greater than or equal to a. Interval notation: open bracket, a, infinity, close parenthesis. Row 4, Inequality: x less than or equal to a. Interval notation: open parenthesis, negative infinity, a, close bracket. Row 5, Inequality: a is less than x is less than b. Interval notation: open parenthesis, a, b, close parenthesis. Row 6, Inequality: a is less than or equal to x is less than b. Interval notation: Open bracket, a, b, close parenthesis. Row 7, Inequality: a is less than x is less than or equal to b. Interval notation: Open parenthesis, a, b, close bracket. Row 8, Inequality: a, less than or equal to x is less than or equal to b. Interval notation: open bracket, a, b, close bracket.\" width=\"975\" height=\"905\" \/><\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/h3>\n<p>Find the domain of the following function: [latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex] .<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\n<p>First identify the input values. The input value is the first coordinate in an <strong>ordered pair<\/strong>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\n<div style=\"text-align: center\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 1<\/h3>\n<p id=\"fs-id1165137852041\">Find the domain of the function:<\/p>\n<p id=\"fs-id1165137466017\" style=\"text-align: center\">[latex]\\left\\{\\left(-5,4\\right),\\left(0,0\\right),\\left(5,-4\\right),\\left(10,-8\\right),\\left(15,-12\\right)\\right\\}[\/latex]<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<div id=\"fs-id1165134225655\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165134355557\"><strong>How To: Given a function written in equation form, find the domain.<\/strong><\/h3>\n<ol id=\"fs-id1165134187286\">\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\n<li>Write the domain in interval form, if possible.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding the Domain of a Function<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)={x}^{2}-1[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135684349\" class=\"solution textbox shaded\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137594433\">The input value, shown by the variable [latex]x[\/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.<\/p>\n<p id=\"fs-id1165135309759\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 2<\/h3>\n<p>Find the domain of the function: [latex]f\\left(x\\right)=5-x+{x}^{3}[\/latex].<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<div id=\"fs-id1165137417188\" class=\"note precalculus howto textbox\">\n<h3>How To:\u00a0Given a function written in an equation form that includes a fraction, find the domain.<\/h3>\n<ol id=\"fs-id1165137463251\">\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for [latex]x[\/latex] . If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\n<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Finding the Domain of a Function Involving a Denominator (Rational Function)<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135641743\" class=\"solution textbox shaded\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137565519\">When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[\/latex].<\/p>\n<div id=\"fs-id1165137736620\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}2-x=0\\hfill \\\\ -x=-2\\hfill \\\\ x=2\\hfill \\end{cases}[\/latex]<\/div>\n<p>Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[\/latex] or [latex]x>2[\/latex]. We can use a symbol known as the union, [latex]\\cup[\/latex], to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<span><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010544\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134036054\">In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  The Domain of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 3<\/h3>\n<p>Find the domain of the function: [latex]f\\left(x\\right)=\\frac{1+4x}{2x - 1}[\/latex].<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<div id=\"fs-id1165135527005\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137733733\">How To: Given a function written in equation form including an even root, find the domain.<\/h3>\n<ol id=\"fs-id1165137820030\">\n<li>Identify the input values.<\/li>\n<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/li>\n<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Finding the Domain of a Function with an Even Root<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137451129\" class=\"solution textbox shaded\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137453224\">When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.<\/p>\n<p id=\"fs-id1165137749755\">Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\n<div id=\"fs-id1165137727831\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}7-x\\ge 0\\hfill \\\\ -x\\ge -7\\hfill \\\\ x\\le 7\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137422794\">Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[\/latex], or [latex]\\left(-\\infty ,7\\right][\/latex].<\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Domain and Range of Square Root Functions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 4<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/precalcone\/chapter\/solutions-2\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<div class=\"\u201ctextbox\u201d textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\n<p id=\"fs-id1165137937737\"><em>Yes. For example, the function [latex]f\\left(x\\right)=-\\frac{1}{\\sqrt{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\n<\/div>\n\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-10836\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, 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\/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><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Ex: The Domain of Rational Functions . <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&#038;feature=youtu.be\">https:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&#038;feature=youtu.be<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Mathispower4u. <strong>Authored by<\/strong>: Ex: Domain and Range of Square Root Functions. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM&#038;feature=youtu.be\">https:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM&#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":276,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"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\":\"copyrighted_video\",\"description\":\"Ex: The Domain of Rational Functions 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