{"id":2029,"date":"2024-07-02T22:30:04","date_gmt":"2024-07-02T22:30:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2029"},"modified":"2025-08-15T13:55:52","modified_gmt":"2025-08-15T13:55:52","slug":"radical-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radical-functions-learn-it-1\/","title":{"raw":"Inverses and Radical Functions: Learn It 1","rendered":"Inverses and Radical Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Learn how to find the inverse (or \"reverse\") of a polynomial function when it's possible<\/li>\r\n \t<li>Figure out how to limit the domain of a polynomial function so you can find its inverse<\/li>\r\n \t<li>Use radical functions to solve real-world problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Radical Functions<\/h2>\r\n<p class=\"whitespace-pre-wrap break-words\">Imagine you're designing a new smartphone app that adjusts screen brightness based on ambient light. You've found that the relationship between light intensity and ideal brightness follows a cubic function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = 0.1x^3 + 1[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">where [latex]x[\/latex] is the light intensity and [latex]f(x)[\/latex] is the ideal brightness setting.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">But what if you want your app to do the reverse \u2013 determine the light intensity when given a brightness setting? This is where <strong>radical functions<\/strong> come into play.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Radical functions are like the \"undo\" button for certain polynomial functions. They allow us to reverse the effects of exponents, helping us solve problems that involve working backwards from a result.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">In our smartphone example, to find the light intensity for a given brightness, we need the inverse of our cubic function. This inverse turns out to be a radical function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f^{-1}(x) = \\sqrt[3]{\\frac{x-1}{0.1}}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">This is a cube root function \u2013 a type of radical function that \"undoes\" the cube in our original polynomial.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>radical functions<\/h3>\r\nA <strong>radical function<\/strong> is a function that involves the use of a radical (root) symbol to indicate the root of a number or expression.\r\n\r\n&nbsp;\r\n\r\nThe general form of a radical function is\r\n<p style=\"text-align: center;\">[latex]f(x) = \\sqrt[n]{g(x)}[\/latex]<\/p>\r\nwhere:\r\n<ul>\r\n \t<li>[latex]\\sqrt[n]{\\cdot}[\/latex] denotes the [latex]n[\/latex]-th root.<\/li>\r\n \t<li>[latex]g(x)[\/latex] is any function of [latex]x[\/latex].<\/li>\r\n \t<li>[latex]n[\/latex] is a positive integer indicating the degree of the root.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Key Points:<\/strong>\r\n<ul>\r\n \t<li><strong>Square Root Function<\/strong>:\r\n<ul>\r\n \t<li>When [latex]n = 2[\/latex], the function is called a square root function. The radical symbol [latex]\\sqrt{\\cdot}[\/latex] is used without explicitly writing the [latex]2[\/latex], known as the \"phantom\" [latex]2[\/latex]. Thus, It is written as [latex]f(x) = \\sqrt{x}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Cube Root Function<\/strong>:\r\n<ul>\r\n \t<li>When [latex]n = 3[\/latex], the function is called a cube root function. It is written as [latex]f(x) = \\sqrt[3]{x}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Higher-Order Root Functions<\/strong>:\r\n<ul>\r\n \t<li>For [latex]n \\ge 3[\/latex], the function represents higher-order roots, such as fourth roots, fifth roots, etc.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">Recall that two functions [latex]f[\/latex] and [latex]g[\/latex] are inverse functions if for every coordinate pair in [latex]f[\/latex], [latex](a, b)[\/latex], there exists a corresponding coordinate pair in the inverse function, [latex]g[\/latex], [latex](b, a)[\/latex]. In other words, the coordinate pairs of the inverse functions have the input and output interchanged. Only one-to-one functions have inverses that are also functions. Recall that a one-to-one function has a unique output value for each input value and passes the horizontal line test.<\/section>While it is not possible to find an inverse function of most polynomial functions, some basic polynomials do have inverses that are functions. Such functions are called <strong>invertible functions<\/strong>, and we use the notation [latex]{f}^{-1}\\left(x\\right)[\/latex].\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Warning<\/strong>: [latex]{f}^{-1}\\left(x\\right)[\/latex] is not the same as the reciprocal of the function [latex]f\\left(x\\right)[\/latex]. This use of [latex]\u20131[\/latex] is reserved to denote inverse functions. To denote the reciprocal of a function [latex]f\\left(x\\right)[\/latex], we would need to write [latex]{\\left(f\\left(x\\right)\\right)}^{-1}=\\frac{1}{f\\left(x\\right)}[\/latex].<\/section>An important relationship between inverse functions is that they \"undo\" each other. If [latex]{f}^{-1}[\/latex] is the inverse of a function [latex]f[\/latex],\u00a0then [latex]f[\/latex]\u00a0is the inverse of the function [latex]{f}^{-1}[\/latex]. In other words, whatever the function [latex]f[\/latex]\u00a0does to [latex]x[\/latex], [latex]{f}^{-1}[\/latex] undoes it\u2014and vice-versa. More formally, we write\r\n<p style=\"text-align: center;\">[latex]{f}^{-1}\\left(f\\left(x\\right)\\right)=x,\\text{for all }x\\text{ in the domain of }f[\/latex]<\/p>\r\nand\r\n<p style=\"text-align: center;\">[latex]f\\left({f}^{-1}\\left(x\\right)\\right)=x,\\text{for all }x\\text{ in the domain of }{f}^{-1}[\/latex]<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial function, find the inverse of the function\u00a0<\/strong>\r\n<ol>\r\n \t<li>Verify that\u00a0[latex]f[\/latex] is a one-to-one function.<\/li>\r\n \t<li>Replace [latex]f\\left(x\\right)[\/latex] with [latex]y[\/latex].<\/li>\r\n \t<li>Interchange [latex]x[\/latex]\u00a0and [latex]y[\/latex].<\/li>\r\n \t<li>Solve for [latex]y[\/latex], and rename the function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the inverse of the function [latex]f\\left(x\\right)=5{x}^{3}+1[\/latex].[reveal-answer q=\"289537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"289537\"]This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=5{x}^{3}+1 \\\\[1mm] x&amp;=5{y}^{3}+1 \\\\[1mm] x - 1&amp;=5{y}^{3} \\\\[1mm] \\dfrac{x - 1}{5}&amp;={y}^{3} \\\\[4mm] {f}^{-1}\\left(x\\right)&amp;=\\sqrt[3]{\\dfrac{x - 1}{5}} \\end{align}[\/latex]<\/p>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nLook at the graph of [latex]f[\/latex] and [latex]{f}^{-1}[\/latex]. Notice that the two graphs are symmetrical about the line [latex]y=x[\/latex]. This is always the case when graphing a function and its inverse function.\r\n\r\nAlso, since the method involved interchanging [latex]x[\/latex]\u00a0and [latex]y[\/latex], notice corresponding points. If [latex]\\left(a,b\\right)[\/latex] is on the graph of [latex]f[\/latex], then [latex]\\left(b,a\\right)[\/latex] is on the graph of [latex]{f}^{-1}[\/latex]. Since [latex]\\left(0,1\\right)[\/latex] is on the graph of [latex]f[\/latex], then [latex]\\left(1,0\\right)[\/latex] is on the graph of [latex]{f}^{-1}[\/latex]. Similarly, since [latex]\\left(1,6\\right)[\/latex] is on the graph of [latex]f[\/latex], then [latex]\\left(6,1\\right)[\/latex] is on the graph of [latex]{f}^{-1}[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02221708\/CNX_Precalc_Figure_03_08_0042.jpg\" alt=\"Graph of f(x)=5x^3+1 and its inverse, f^(-1)(x)=3sqrt((x-1)\/(5)).\" width=\"487\" height=\"554\" \/> Graph of f(x) and its inverse f^(-1)(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24669[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Learn how to find the inverse (or &#8220;reverse&#8221;) of a polynomial function when it&#8217;s possible<\/li>\n<li>Figure out how to limit the domain of a polynomial function so you can find its inverse<\/li>\n<li>Use radical functions to solve real-world problems<\/li>\n<\/ul>\n<\/section>\n<h2>Radical Functions<\/h2>\n<p class=\"whitespace-pre-wrap break-words\">Imagine you&#8217;re designing a new smartphone app that adjusts screen brightness based on ambient light. You&#8217;ve found that the relationship between light intensity and ideal brightness follows a cubic function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = 0.1x^3 + 1[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">where [latex]x[\/latex] is the light intensity and [latex]f(x)[\/latex] is the ideal brightness setting.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">But what if you want your app to do the reverse \u2013 determine the light intensity when given a brightness setting? This is where <strong>radical functions<\/strong> come into play.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Radical functions are like the &#8220;undo&#8221; button for certain polynomial functions. They allow us to reverse the effects of exponents, helping us solve problems that involve working backwards from a result.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">In our smartphone example, to find the light intensity for a given brightness, we need the inverse of our cubic function. This inverse turns out to be a radical function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f^{-1}(x) = \\sqrt[3]{\\frac{x-1}{0.1}}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">This is a cube root function \u2013 a type of radical function that &#8220;undoes&#8221; the cube in our original polynomial.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>radical functions<\/h3>\n<p>A <strong>radical function<\/strong> is a function that involves the use of a radical (root) symbol to indicate the root of a number or expression.<\/p>\n<p>&nbsp;<\/p>\n<p>The general form of a radical function is<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = \\sqrt[n]{g(x)}[\/latex]<\/p>\n<p>where:<\/p>\n<ul>\n<li>[latex]\\sqrt[n]{\\cdot}[\/latex] denotes the [latex]n[\/latex]-th root.<\/li>\n<li>[latex]g(x)[\/latex] is any function of [latex]x[\/latex].<\/li>\n<li>[latex]n[\/latex] is a positive integer indicating the degree of the root.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Key Points:<\/strong><\/p>\n<ul>\n<li><strong>Square Root Function<\/strong>:\n<ul>\n<li>When [latex]n = 2[\/latex], the function is called a square root function. The radical symbol [latex]\\sqrt{\\cdot}[\/latex] is used without explicitly writing the [latex]2[\/latex], known as the &#8220;phantom&#8221; [latex]2[\/latex]. Thus, It is written as [latex]f(x) = \\sqrt{x}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Cube Root Function<\/strong>:\n<ul>\n<li>When [latex]n = 3[\/latex], the function is called a cube root function. It is written as [latex]f(x) = \\sqrt[3]{x}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Higher-Order Root Functions<\/strong>:\n<ul>\n<li>For [latex]n \\ge 3[\/latex], the function represents higher-order roots, such as fourth roots, fifth roots, etc.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">Recall that two functions [latex]f[\/latex] and [latex]g[\/latex] are inverse functions if for every coordinate pair in [latex]f[\/latex], [latex](a, b)[\/latex], there exists a corresponding coordinate pair in the inverse function, [latex]g[\/latex], [latex](b, a)[\/latex]. In other words, the coordinate pairs of the inverse functions have the input and output interchanged. Only one-to-one functions have inverses that are also functions. Recall that a one-to-one function has a unique output value for each input value and passes the horizontal line test.<\/section>\n<p>While it is not possible to find an inverse function of most polynomial functions, some basic polynomials do have inverses that are functions. Such functions are called <strong>invertible functions<\/strong>, and we use the notation [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Warning<\/strong>: [latex]{f}^{-1}\\left(x\\right)[\/latex] is not the same as the reciprocal of the function [latex]f\\left(x\\right)[\/latex]. This use of [latex]\u20131[\/latex] is reserved to denote inverse functions. To denote the reciprocal of a function [latex]f\\left(x\\right)[\/latex], we would need to write [latex]{\\left(f\\left(x\\right)\\right)}^{-1}=\\frac{1}{f\\left(x\\right)}[\/latex].<\/section>\n<p>An important relationship between inverse functions is that they &#8220;undo&#8221; each other. If [latex]{f}^{-1}[\/latex] is the inverse of a function [latex]f[\/latex],\u00a0then [latex]f[\/latex]\u00a0is the inverse of the function [latex]{f}^{-1}[\/latex]. In other words, whatever the function [latex]f[\/latex]\u00a0does to [latex]x[\/latex], [latex]{f}^{-1}[\/latex] undoes it\u2014and vice-versa. More formally, we write<\/p>\n<p style=\"text-align: center;\">[latex]{f}^{-1}\\left(f\\left(x\\right)\\right)=x,\\text{for all }x\\text{ in the domain of }f[\/latex]<\/p>\n<p>and<\/p>\n<p style=\"text-align: center;\">[latex]f\\left({f}^{-1}\\left(x\\right)\\right)=x,\\text{for all }x\\text{ in the domain of }{f}^{-1}[\/latex]<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial function, find the inverse of the function\u00a0<\/strong><\/p>\n<ol>\n<li>Verify that\u00a0[latex]f[\/latex] is a one-to-one function.<\/li>\n<li>Replace [latex]f\\left(x\\right)[\/latex] with [latex]y[\/latex].<\/li>\n<li>Interchange [latex]x[\/latex]\u00a0and [latex]y[\/latex].<\/li>\n<li>Solve for [latex]y[\/latex], and rename the function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse of the function [latex]f\\left(x\\right)=5{x}^{3}+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q289537\">Show Solution<\/button><\/p>\n<div id=\"q289537\" class=\"hidden-answer\" style=\"display: none\">This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=5{x}^{3}+1 \\\\[1mm] x&=5{y}^{3}+1 \\\\[1mm] x - 1&=5{y}^{3} \\\\[1mm] \\dfrac{x - 1}{5}&={y}^{3} \\\\[4mm] {f}^{-1}\\left(x\\right)&=\\sqrt[3]{\\dfrac{x - 1}{5}} \\end{align}[\/latex]<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Look at the graph of [latex]f[\/latex] and [latex]{f}^{-1}[\/latex]. Notice that the two graphs are symmetrical about the line [latex]y=x[\/latex]. This is always the case when graphing a function and its inverse function.<\/p>\n<p>Also, since the method involved interchanging [latex]x[\/latex]\u00a0and [latex]y[\/latex], notice corresponding points. If [latex]\\left(a,b\\right)[\/latex] is on the graph of [latex]f[\/latex], then [latex]\\left(b,a\\right)[\/latex] is on the graph of [latex]{f}^{-1}[\/latex]. Since [latex]\\left(0,1\\right)[\/latex] is on the graph of [latex]f[\/latex], then [latex]\\left(1,0\\right)[\/latex] is on the graph of [latex]{f}^{-1}[\/latex]. Similarly, since [latex]\\left(1,6\\right)[\/latex] is on the graph of [latex]f[\/latex], then [latex]\\left(6,1\\right)[\/latex] is on the graph of [latex]{f}^{-1}[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02221708\/CNX_Precalc_Figure_03_08_0042.jpg\" alt=\"Graph of f(x)=5x^3+1 and its inverse, f^(-1)(x)=3sqrt((x-1)\/(5)).\" width=\"487\" height=\"554\" \/><figcaption class=\"wp-caption-text\">Graph of f(x) and its inverse f^(-1)(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24669\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24669&theme=lumen&iframe_resize_id=ohm24669&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":232,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2029"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2029\/revisions"}],"predecessor-version":[{"id":7794,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2029\/revisions\/7794"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/232"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2029\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2029"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2029"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2029"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}