{"id":5034,"date":"2017-12-31T00:40:50","date_gmt":"2017-12-31T00:40:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=5034"},"modified":"2023-11-08T13:20:22","modified_gmt":"2023-11-08T13:20:22","slug":"radical-functions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/radical-functions\/","title":{"raw":"9.3 - Radical Functions","rendered":"9.3 &#8211; Radical Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>(9.3.1) - Evaluating Radical functions<\/li>\r\n \t<li>(9.3.2) - Finding the domain of a radical function<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a\u00a0<strong>radical function<\/strong>.\r\n\r\nThe square root function is [latex]f(x) = \\sqrt{x}[\/latex].\r\n\r\nThe cube root function is\u00a0[latex]f(x) = \\sqrt[3]{x}[\/latex].\r\n<div class=\"textbox shaded\">A\u00a0<strong>radical function<\/strong>\u00a0is a function that is defined by a radical expression.<\/div>\r\nThe following are examples of rational functions: [latex]f(x)=\\sqrt{2x^4-5}[\/latex];\u00a0[latex]g(x)=\\sqrt[3]{4x-7}[\/latex];\u00a0[latex]h(x)=\\sqrt[7]{-8x^2+4}[\/latex].\r\n<h1>(9.3.1) - Evaluating a radical function<\/h1>\r\nTo evaluate a radical function, we find the value of [latex]f(x)[\/latex] for a given value of [latex]x[\/latex]\u00a0just as we did in our previous work with functions.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nFor the function [latex]f(x) = \\sqrt{2x-1}[\/latex], find [latex]f(5)[\/latex]\r\n\r\n[reveal-answer q=\"505616\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"505616\"]\r\n\r\nTo evaluate\u00a0[latex]f(5)[\/latex], substitute 5 for\u00a0[latex]x[\/latex]:\r\n<p style=\"text-align: center\">[latex]f(5)=\\sqrt{2 \\cdot 5-1}[\/latex]<\/p>\r\nSimplify:\r\n<p style=\"text-align: center\">[latex]f(5) = \\sqrt{9}[\/latex]<\/p>\r\nTake the square root:\r\n<p style=\"text-align: center\">[latex]3[\/latex]<\/p>\r\n<strong>Answer<\/strong>\r\n\r\n[latex]3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nFor the function [latex]f(x) = \\sqrt{-4x+5}[\/latex], find [latex]f(-5)[\/latex]\r\n\r\n[reveal-answer q=\"179741\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"179741\"]\r\n\r\nTo evaluate\u00a0[latex]f(5)[\/latex], substitute 5 for\u00a0[latex]x[\/latex]:\r\n<p style=\"text-align: center\">[latex]f(-5)=\\sqrt{-4 \\cdot (-5)+5}[\/latex]<\/p>\r\nSimplify:\r\n<p style=\"text-align: center\">[latex]f(5) = \\sqrt{25}[\/latex]<\/p>\r\nTake the square root:\r\n<p style=\"text-align: center\">[latex]5[\/latex]<\/p>\r\n<strong> Answer <\/strong>\r\n\r\n[latex]5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]73906[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following example we evaluate a cube root function.\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMPLE<\/h3>\r\nFor the function [latex]f(x) = \\sqrt[3]{x-29}[\/latex], find [latex]f(2)[\/latex]\r\n\r\n[reveal-answer q=\"544966\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"544966\"]\r\n\r\nTo evaluate\u00a0[latex]f(2)[\/latex], substitute [latex]2[\/latex] for\u00a0[latex]x[\/latex]:\r\n\r\n[latex]f(2)=\\sqrt{2-29}[\/latex]\r\n\r\nSimplify:\r\n\r\n[latex]f(5) = \\sqrt[3]{-27}[\/latex]\r\n\r\nTake the square root:\r\n\r\n[latex]-3[\/latex]\r\n\r\n<strong> Answer <\/strong>\r\n\r\n[latex]-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h1>(9.3.2) - Finding the domain of a radical function<\/h1>\r\nFor the <strong>square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex]. The following is a graph of the square root function:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193620\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/>\r\n\r\nFor the <strong>cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). Here is the graph of the cube root function:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193622\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/>\r\n\r\nWe use this to find the domains of other radical functions.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function written in equation form including an even root, find the domain.<\/h3>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/li>\r\n \t<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Function with an Even Root<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].\r\n\r\n[reveal-answer q=\"722013\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"722013\"]\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\r\n<div class=\"textbox key-takeaways\">\r\n<h3>EXAMPLE<\/h3>\r\nFind the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].\r\n\r\n[reveal-answer q=\"643325\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"643325\"][latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=30831&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92940&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain and Range<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].\r\n\r\n[reveal-answer q=\"605324\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"605324\"]\r\nWe cannot take the square root of a negative number, so the value inside the radical must be nonnegative.\r\n\r\n[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]\r\n\r\nThe domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].\r\n\r\nWe then find the range. We know that [latex]f\\left(-4\\right)=0[\/latex], and the function value increases as [latex]x[\/latex] increases without any upper limit. We conclude that the range of [latex]f[\/latex] is [latex]\\left[0,\\infty \\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph below represents the function [latex]f[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193624\/CNX_Precalc_Figure_01_02_0202.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>(9.3.1) &#8211; Evaluating Radical functions<\/li>\n<li>(9.3.2) &#8211; Finding the domain of a radical function<\/li>\n<\/ul>\n<\/div>\n<p>In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a\u00a0<strong>radical function<\/strong>.<\/p>\n<p>The square root function is [latex]f(x) = \\sqrt{x}[\/latex].<\/p>\n<p>The cube root function is\u00a0[latex]f(x) = \\sqrt[3]{x}[\/latex].<\/p>\n<div class=\"textbox shaded\">A\u00a0<strong>radical function<\/strong>\u00a0is a function that is defined by a radical expression.<\/div>\n<p>The following are examples of rational functions: [latex]f(x)=\\sqrt{2x^4-5}[\/latex];\u00a0[latex]g(x)=\\sqrt[3]{4x-7}[\/latex];\u00a0[latex]h(x)=\\sqrt[7]{-8x^2+4}[\/latex].<\/p>\n<h1>(9.3.1) &#8211; Evaluating a radical function<\/h1>\n<p>To evaluate a radical function, we find the value of [latex]f(x)[\/latex] for a given value of [latex]x[\/latex]\u00a0just as we did in our previous work with functions.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>For the function [latex]f(x) = \\sqrt{2x-1}[\/latex], find [latex]f(5)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q505616\">Show Solution<\/span><\/p>\n<div id=\"q505616\" class=\"hidden-answer\" style=\"display: none\">\n<p>To evaluate\u00a0[latex]f(5)[\/latex], substitute 5 for\u00a0[latex]x[\/latex]:<\/p>\n<p style=\"text-align: center\">[latex]f(5)=\\sqrt{2 \\cdot 5-1}[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p style=\"text-align: center\">[latex]f(5) = \\sqrt{9}[\/latex]<\/p>\n<p>Take the square root:<\/p>\n<p style=\"text-align: center\">[latex]3[\/latex]<\/p>\n<p><strong>Answer<\/strong><\/p>\n<p>[latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>For the function [latex]f(x) = \\sqrt{-4x+5}[\/latex], find [latex]f(-5)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q179741\">Show Solution<\/span><\/p>\n<div id=\"q179741\" class=\"hidden-answer\" style=\"display: none\">\n<p>To evaluate\u00a0[latex]f(5)[\/latex], substitute 5 for\u00a0[latex]x[\/latex]:<\/p>\n<p style=\"text-align: center\">[latex]f(-5)=\\sqrt{-4 \\cdot (-5)+5}[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p style=\"text-align: center\">[latex]f(5) = \\sqrt{25}[\/latex]<\/p>\n<p>Take the square root:<\/p>\n<p style=\"text-align: center\">[latex]5[\/latex]<\/p>\n<p><strong> Answer <\/strong><\/p>\n<p>[latex]5[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm73906\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=73906&theme=oea&iframe_resize_id=ohm73906&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following example we evaluate a cube root function.<\/p>\n<div class=\"textbox exercises\">\n<h3>ExAMPLE<\/h3>\n<p>For the function [latex]f(x) = \\sqrt[3]{x-29}[\/latex], find [latex]f(2)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q544966\">Show Solution<\/span><\/p>\n<div id=\"q544966\" class=\"hidden-answer\" style=\"display: none\">\n<p>To evaluate\u00a0[latex]f(2)[\/latex], substitute [latex]2[\/latex] for\u00a0[latex]x[\/latex]:<\/p>\n<p>[latex]f(2)=\\sqrt{2-29}[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p>[latex]f(5) = \\sqrt[3]{-27}[\/latex]<\/p>\n<p>Take the square root:<\/p>\n<p>[latex]-3[\/latex]<\/p>\n<p><strong> Answer <\/strong><\/p>\n<p>[latex]-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1>(9.3.2) &#8211; Finding the domain of a radical function<\/h1>\n<p>For the <strong>square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex]. The following is a graph of the square root function:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193620\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/><\/p>\n<p>For the <strong>cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). Here is the graph of the cube root function:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193622\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/><\/p>\n<p>We use this to find the domains of other radical functions.<\/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: Finding the Domain of a Function with an Even Root<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q722013\">Solution<\/span><\/p>\n<div id=\"q722013\" class=\"hidden-answer\" style=\"display: none\">\nWhen 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-1\" title=\"Ex: Domain and Range of Square Root Functions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>EXAMPLE<\/h3>\n<p>Find the domain of the function [latex]f\\left(x\\right)=\\sqrt{5+2x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q643325\">Solution<\/span><\/p>\n<div id=\"q643325\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left[-\\frac{5}{2},\\infty \\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=30831&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92940&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain and Range<\/h3>\n<p>Find the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q605324\">Solution<\/span><\/p>\n<div id=\"q605324\" class=\"hidden-answer\" style=\"display: none\">\nWe cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\n<p>[latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/p>\n<p>The domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].<\/p>\n<p>We then find the range. We know that [latex]f\\left(-4\\right)=0[\/latex], and the function value increases as [latex]x[\/latex] increases without any upper limit. We conclude that the range of [latex]f[\/latex] is [latex]\\left[0,\\infty \\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph below represents the function [latex]f[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193624\/CNX_Precalc_Figure_01_02_0202.jpg\" alt=\"Graph of a square root function at (-4, 0).\" width=\"487\" height=\"330\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n\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-5034\">\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>Question ID#73906. <strong>Authored by<\/strong>: Meacham,William. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><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>Intermediate Algebra. <strong>Authored by<\/strong>: Lynn Marecek et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/02776133-d49d-49cb-bfaa-67c7f61b25a1@3.16\">http:\/\/cnx.org\/contents\/02776133-d49d-49cb-bfaa-67c7f61b25a1@3.16<\/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\/02776133-d49d-49cb-bfaa-67c7f61b25a1@3.16<\/li><li>Question ID 30831, 32883. <strong>Authored by<\/strong>: Smart, Jim. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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