{"id":147,"date":"2023-06-21T13:22:38","date_gmt":"2023-06-21T13:22:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/domain-of-a-composition\/"},"modified":"2023-10-19T22:37:50","modified_gmt":"2023-10-19T22:37:50","slug":"domain-of-a-composition","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/domain-of-a-composition\/","title":{"raw":"\u25aa   Domain of a Composition","rendered":"\u25aa   Domain of a Composition"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the domain of a composite function.<\/li>\r\n \t<li>Decompose a composite function.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs we discussed previously, the <strong>domain of a composite function<\/strong> such as [latex]f\\circ g[\/latex] is dependent on the domain of [latex]g[\/latex] and the domain of [latex]f[\/latex]. It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as [latex]f\\circ g[\/latex]. Let us assume we know the domains of the functions [latex]f[\/latex] and [latex]g[\/latex] separately. If we write the composite function for an input [latex]x[\/latex] as [latex]f\\left(g\\left(x\\right)\\right)[\/latex], we can see right away that [latex]x[\/latex] must be a member of the domain of [latex]g[\/latex] in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that [latex]g\\left(x\\right)[\/latex] must be a member of the domain of [latex]f[\/latex], otherwise the second function evaluation in [latex]f\\left(g\\left(x\\right)\\right)[\/latex] cannot be completed, and the expression is still undefined. Thus the domain of [latex]f\\circ g[\/latex] consists of only those inputs in the domain of [latex]g[\/latex] that produce outputs from [latex]g[\/latex] belonging to the domain of [latex]f[\/latex]. Note that the domain of [latex]f[\/latex] composed with [latex]g[\/latex] is the set of all [latex]x[\/latex] such that [latex]x[\/latex] is in the domain of [latex]g[\/latex] and [latex]g\\left(x\\right)[\/latex] is in the domain of [latex]f[\/latex].\r\n<div class=\"textbox\">\r\n<h3>A General Note: Domain of a Composite Function<\/h3>\r\nThe domain of a composite function [latex]f\\left(g\\left(x\\right)\\right)[\/latex] is the set of those inputs [latex]x[\/latex] in the domain of [latex]g[\/latex] for which [latex]g\\left(x\\right)[\/latex] is in the domain of [latex]f[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function composition [latex]f\\left(g\\left(x\\right)\\right)[\/latex], determine its domain.<\/h3>\r\n<ol>\r\n \t<li>Find the domain of [latex]g[\/latex].<\/li>\r\n \t<li>Find the domain of [latex]f[\/latex].<\/li>\r\n \t<li>Find those inputs,\u00a0[latex]x[\/latex],\u00a0in the domain of [latex]g[\/latex]\u00a0for which [latex]g(x)[\/latex]\u00a0is in the domain of [latex]f[\/latex]. That is, exclude those inputs, [latex]x[\/latex], from the domain of [latex]g[\/latex]\u00a0for which [latex]g(x)[\/latex]\u00a0is not in the domain of [latex]f[\/latex]. The resulting set is the domain of [latex]f\\circ g[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Composite Function<\/h3>\r\nFind the domain of\r\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)\\text{ where}f\\left(x\\right)=\\dfrac{5}{x - 1}\\text{ and }g\\left(x\\right)=\\dfrac{4}{3x - 2}[\/latex]<\/p>\r\n[reveal-answer q=\"924127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"924127\"]\r\n\r\nThe domain of [latex]g\\left(x\\right)[\/latex] consists of all real numbers except [latex]x=\\frac{2}{3}[\/latex], since that input value would cause us to divide by 0. Likewise, the domain of [latex]f[\/latex] consists of all real numbers except 1. So we need to exclude from the domain of [latex]g\\left(x\\right)[\/latex] that value of [latex]x[\/latex] for which [latex]g\\left(x\\right)=1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\dfrac{4}{3x - 2}=1\\hspace{5mm}&amp;&amp;\\text{Set}\\hspace{2mm}g(x)\\hspace{2mm}\\text{equal to 1} \\\\[2mm]&amp; 4=3x - 2 &amp;&amp;\\text{Multiply by}\\hspace{2mm} 3x-2\\\\[2mm]&amp; 6=3x&amp;&amp;\\text{Add 2 to both sides}\\\\[2mm]&amp; x=2&amp;&amp;\\text{Divide by 3} \\end{align}[\/latex]<\/p>\r\nSo the domain of [latex]f\\circ g[\/latex] is the set of all real numbers except [latex]\\frac{2}{3}[\/latex] and [latex]2[\/latex]. This means that\r\n<p style=\"text-align: center;\">[latex]x\\ne \\frac{2}{3}\\hspace{2mm}\\text{or}\\hspace{2mm}x\\ne 2[\/latex]<\/p>\r\nWe can write this in interval notation as\r\n<p style=\"text-align: center;\">[latex]\\left(-\\infty ,\\frac{2}{3}\\right)\\cup \\left(\\frac{2}{3},2\\right)\\cup \\left(2,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Composite Function Involving Radicals<\/h3>\r\nFind the domain of\r\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)\\text{ where}f\\left(x\\right)=\\sqrt{x+2}\\text{ and }g\\left(x\\right)=\\sqrt{3-x}[\/latex]<\/p>\r\n[reveal-answer q=\"880067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880067\"]\r\n\r\nBecause we cannot take the square root of a negative number, the domain of [latex]g[\/latex] is [latex]\\left(-\\infty ,3\\right][\/latex]. Now we check the domain of the composite function\r\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\sqrt{\\sqrt{3-x}+2}[\/latex]<\/p>\r\nFor [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex], we need [latex]\\sqrt{3-x}+2\\ge{0}[\/latex], since the radicand of a square root must be positive. Since square roots are positive, [latex]\\sqrt{3-x}\\ge{0}[\/latex], or [latex]3-x\\ge{0}[\/latex], which gives a domain of [latex]\\left(f\\circ g\\right)\\left(x\\right) = (-\\infty,3][\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nThis example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of [latex]f\\circ g[\/latex] can contain values that are not in the domain of [latex]f[\/latex], though they must be in the domain of [latex]g[\/latex].\r\n\r\nYou cannot rely on an algorithm to find the domain of a composite function. Rather, you will need to first ask yourself \"what is the domain of the inner function\", and determine whether this set will comply with the domain restrictions of the outer function. In this case, the set [latex](-\\infty,3][\/latex] ensures a non-negative output for the inner function, which will in turn ensure a positive input for the composite function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the domain of\r\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)\\text{ where}f\\left(x\\right)=\\dfrac{1}{x - 2}\\text{ and }g\\left(x\\right)=\\sqrt{x+4}[\/latex]<\/p>\r\n[reveal-answer q=\"936114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"936114\"]\r\n\r\n[latex]\\left[-4,0\\right)\\cup \\left(0,\\infty \\right)[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]1603[\/ohm_question]\r\n\r\n[ohm_question]1604[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWe can use graphs to visualize the domain that results from a composition of two functions.\r\nGraph the two functions below with an online graphing calculator.\r\n<ol>\r\n \t<li>[latex]f(x)=\\sqrt{3-x}[\/latex]<\/li>\r\n \t<li>[latex]g(t) = \\sqrt{x+4}[\/latex]<\/li>\r\n<\/ol>\r\nNext, create a new function, [latex]h(x) = g(f(x))[\/latex]. \u00a0Based on the graph, what is the domain of this function? Explain why [latex]g(f(x))[\/latex] and [latex]f(x)[\/latex] have the same domain.\r\n\r\nNow define another composition, [latex]p(x) = f(g(x)[\/latex]. \u00a0What is the domain of this function? Explain why you can evaluate\u00a0[latex]g(10)[\/latex], but not [latex]p(10)[\/latex].\r\n\r\n<\/div>\r\n<h2>Decompose a Composite Function<\/h2>\r\nIn some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There is almost always more than one way to <strong>decompose a composite function<\/strong>, so we may choose the decomposition that appears to be most obvious.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Decomposing a Function<\/h3>\r\nWrite [latex]f\\left(x\\right)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.\r\n\r\n[reveal-answer q=\"702975\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"702975\"]\r\n\r\nWe are looking for two functions, [latex]g[\/latex] and [latex]h[\/latex], so [latex]f\\left(x\\right)=g\\left(h\\left(x\\right)\\right)[\/latex]. To do this, we look for a function inside a function in the formula for [latex]f\\left(x\\right)[\/latex]. As one possibility, we might notice that the expression [latex]5-{x}^{2}[\/latex] is the inside of the square root. We could then decompose the function as\r\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)=5-{x}^{2}\\hspace{2mm}\\text{and}\\hspace{2mm}g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\nWe can check our answer by recomposing the functions.\r\n<p style=\"text-align: center;\">[latex]g\\left(h\\left(x\\right)\\right)=g\\left(5-{x}^{2}\\right)=\\sqrt{5-{x}^{2}}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nFor every composition there are infinitely many possible function pairs that will work. In this case, another function pair where\u00a0[latex]g\\left(h\\left(x\\right)\\right)=\\sqrt{5-{x}^{2}}[\/latex]\u00a0 is\u00a0 [latex]h(x)=x^2[\/latex] and [latex]g(x)=\\sqrt{5-x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]f\\left(x\\right)=\\dfrac{4}{3-\\sqrt{4+{x}^{2}}}[\/latex] as the composition of two functions.\r\n\r\n[reveal-answer q=\"489928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489928\"]\r\n\r\nPossible answer:\r\n<p id=\"fs-id1165135333608\">[latex]g\\left(x\\right)=\\sqrt{4+{x}^{2}}[\/latex]<\/p>\r\n[latex]h\\left(x\\right)=\\dfrac{4}{3-x}[\/latex]\r\n\r\n[latex]f=h\\circ g[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]32917[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the domain of a composite function.<\/li>\n<li>Decompose a composite function.<\/li>\n<\/ul>\n<\/div>\n<p>As we discussed previously, the <strong>domain of a composite function<\/strong> such as [latex]f\\circ g[\/latex] is dependent on the domain of [latex]g[\/latex] and the domain of [latex]f[\/latex]. It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as [latex]f\\circ g[\/latex]. Let us assume we know the domains of the functions [latex]f[\/latex] and [latex]g[\/latex] separately. If we write the composite function for an input [latex]x[\/latex] as [latex]f\\left(g\\left(x\\right)\\right)[\/latex], we can see right away that [latex]x[\/latex] must be a member of the domain of [latex]g[\/latex] in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that [latex]g\\left(x\\right)[\/latex] must be a member of the domain of [latex]f[\/latex], otherwise the second function evaluation in [latex]f\\left(g\\left(x\\right)\\right)[\/latex] cannot be completed, and the expression is still undefined. Thus the domain of [latex]f\\circ g[\/latex] consists of only those inputs in the domain of [latex]g[\/latex] that produce outputs from [latex]g[\/latex] belonging to the domain of [latex]f[\/latex]. Note that the domain of [latex]f[\/latex] composed with [latex]g[\/latex] is the set of all [latex]x[\/latex] such that [latex]x[\/latex] is in the domain of [latex]g[\/latex] and [latex]g\\left(x\\right)[\/latex] is in the domain of [latex]f[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Domain of a Composite Function<\/h3>\n<p>The domain of a composite function [latex]f\\left(g\\left(x\\right)\\right)[\/latex] is the set of those inputs [latex]x[\/latex] in the domain of [latex]g[\/latex] for which [latex]g\\left(x\\right)[\/latex] is in the domain of [latex]f[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function composition [latex]f\\left(g\\left(x\\right)\\right)[\/latex], determine its domain.<\/h3>\n<ol>\n<li>Find the domain of [latex]g[\/latex].<\/li>\n<li>Find the domain of [latex]f[\/latex].<\/li>\n<li>Find those inputs,\u00a0[latex]x[\/latex],\u00a0in the domain of [latex]g[\/latex]\u00a0for which [latex]g(x)[\/latex]\u00a0is in the domain of [latex]f[\/latex]. That is, exclude those inputs, [latex]x[\/latex], from the domain of [latex]g[\/latex]\u00a0for which [latex]g(x)[\/latex]\u00a0is not in the domain of [latex]f[\/latex]. The resulting set is the domain of [latex]f\\circ g[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Composite Function<\/h3>\n<p>Find the domain of<\/p>\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)\\text{ where}f\\left(x\\right)=\\dfrac{5}{x - 1}\\text{ and }g\\left(x\\right)=\\dfrac{4}{3x - 2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924127\">Show Solution<\/span><\/p>\n<div id=\"q924127\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain of [latex]g\\left(x\\right)[\/latex] consists of all real numbers except [latex]x=\\frac{2}{3}[\/latex], since that input value would cause us to divide by 0. Likewise, the domain of [latex]f[\/latex] consists of all real numbers except 1. So we need to exclude from the domain of [latex]g\\left(x\\right)[\/latex] that value of [latex]x[\/latex] for which [latex]g\\left(x\\right)=1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\dfrac{4}{3x - 2}=1\\hspace{5mm}&&\\text{Set}\\hspace{2mm}g(x)\\hspace{2mm}\\text{equal to 1} \\\\[2mm]& 4=3x - 2 &&\\text{Multiply by}\\hspace{2mm} 3x-2\\\\[2mm]& 6=3x&&\\text{Add 2 to both sides}\\\\[2mm]& x=2&&\\text{Divide by 3} \\end{align}[\/latex]<\/p>\n<p>So the domain of [latex]f\\circ g[\/latex] is the set of all real numbers except [latex]\\frac{2}{3}[\/latex] and [latex]2[\/latex]. This means that<\/p>\n<p style=\"text-align: center;\">[latex]x\\ne \\frac{2}{3}\\hspace{2mm}\\text{or}\\hspace{2mm}x\\ne 2[\/latex]<\/p>\n<p>We can write this in interval notation as<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-\\infty ,\\frac{2}{3}\\right)\\cup \\left(\\frac{2}{3},2\\right)\\cup \\left(2,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Composite Function Involving Radicals<\/h3>\n<p>Find the domain of<\/p>\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)\\text{ where}f\\left(x\\right)=\\sqrt{x+2}\\text{ and }g\\left(x\\right)=\\sqrt{3-x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880067\">Show Solution<\/span><\/p>\n<div id=\"q880067\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because we cannot take the square root of a negative number, the domain of [latex]g[\/latex] is [latex]\\left(-\\infty ,3\\right][\/latex]. Now we check the domain of the composite function<\/p>\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)=\\sqrt{\\sqrt{3-x}+2}[\/latex]<\/p>\n<p>For [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex], we need [latex]\\sqrt{3-x}+2\\ge{0}[\/latex], since the radicand of a square root must be positive. Since square roots are positive, [latex]\\sqrt{3-x}\\ge{0}[\/latex], or [latex]3-x\\ge{0}[\/latex], which gives a domain of [latex]\\left(f\\circ g\\right)\\left(x\\right) = (-\\infty,3][\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of [latex]f\\circ g[\/latex] can contain values that are not in the domain of [latex]f[\/latex], though they must be in the domain of [latex]g[\/latex].<\/p>\n<p>You cannot rely on an algorithm to find the domain of a composite function. Rather, you will need to first ask yourself &#8220;what is the domain of the inner function&#8221;, and determine whether this set will comply with the domain restrictions of the outer function. In this case, the set [latex](-\\infty,3][\/latex] ensures a non-negative output for the inner function, which will in turn ensure a positive input for the composite function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the domain of<\/p>\n<p style=\"text-align: center;\">[latex]\\left(f\\circ g\\right)\\left(x\\right)\\text{ where}f\\left(x\\right)=\\dfrac{1}{x - 2}\\text{ and }g\\left(x\\right)=\\sqrt{x+4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q936114\">Show Solution<\/span><\/p>\n<div id=\"q936114\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-4,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm1603\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1603&theme=oea&iframe_resize_id=ohm1603&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm1604\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1604&theme=oea&iframe_resize_id=ohm1604&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>We can use graphs to visualize the domain that results from a composition of two functions.<br \/>\nGraph the two functions below with an online graphing calculator.<\/p>\n<ol>\n<li>[latex]f(x)=\\sqrt{3-x}[\/latex]<\/li>\n<li>[latex]g(t) = \\sqrt{x+4}[\/latex]<\/li>\n<\/ol>\n<p>Next, create a new function, [latex]h(x) = g(f(x))[\/latex]. \u00a0Based on the graph, what is the domain of this function? Explain why [latex]g(f(x))[\/latex] and [latex]f(x)[\/latex] have the same domain.<\/p>\n<p>Now define another composition, [latex]p(x) = f(g(x)[\/latex]. \u00a0What is the domain of this function? Explain why you can evaluate\u00a0[latex]g(10)[\/latex], but not [latex]p(10)[\/latex].<\/p>\n<\/div>\n<h2>Decompose a Composite Function<\/h2>\n<p>In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There is almost always more than one way to <strong>decompose a composite function<\/strong>, so we may choose the decomposition that appears to be most obvious.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Decomposing a Function<\/h3>\n<p>Write [latex]f\\left(x\\right)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q702975\">Show Solution<\/span><\/p>\n<div id=\"q702975\" class=\"hidden-answer\" style=\"display: none\">\n<p>We are looking for two functions, [latex]g[\/latex] and [latex]h[\/latex], so [latex]f\\left(x\\right)=g\\left(h\\left(x\\right)\\right)[\/latex]. To do this, we look for a function inside a function in the formula for [latex]f\\left(x\\right)[\/latex]. As one possibility, we might notice that the expression [latex]5-{x}^{2}[\/latex] is the inside of the square root. We could then decompose the function as<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)=5-{x}^{2}\\hspace{2mm}\\text{and}\\hspace{2mm}g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<p>We can check our answer by recomposing the functions.<\/p>\n<p style=\"text-align: center;\">[latex]g\\left(h\\left(x\\right)\\right)=g\\left(5-{x}^{2}\\right)=\\sqrt{5-{x}^{2}}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>For every composition there are infinitely many possible function pairs that will work. In this case, another function pair where\u00a0[latex]g\\left(h\\left(x\\right)\\right)=\\sqrt{5-{x}^{2}}[\/latex]\u00a0 is\u00a0 [latex]h(x)=x^2[\/latex] and [latex]g(x)=\\sqrt{5-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]f\\left(x\\right)=\\dfrac{4}{3-\\sqrt{4+{x}^{2}}}[\/latex] as the composition of two functions.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q489928\">Show Solution<\/span><\/p>\n<div id=\"q489928\" class=\"hidden-answer\" style=\"display: none\">\n<p>Possible answer:<\/p>\n<p id=\"fs-id1165135333608\">[latex]g\\left(x\\right)=\\sqrt{4+{x}^{2}}[\/latex]<\/p>\n<p>[latex]h\\left(x\\right)=\\dfrac{4}{3-x}[\/latex]<\/p>\n<p>[latex]f=h\\circ g[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm32917\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32917&theme=oea&iframe_resize_id=ohm32917&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/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-147\">\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>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 15772. <strong>Authored by<\/strong>: Master,Course. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1600. <strong>Authored by<\/strong>: WebWork-Rochester, mb Lippman,David. <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>: IMathAS Community License CC-BY + GPL<\/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":395986,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 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