{"id":1131,"date":"2024-04-11T16:50:38","date_gmt":"2024-04-11T16:50:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1131"},"modified":"2024-08-04T11:32:57","modified_gmt":"2024-08-04T11:32:57","slug":"inverse-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/inverse-functions-fresh-take\/","title":{"raw":"Inverse Functions: Fresh Take","rendered":"Inverse Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify conditions for a function's inverse using the horizontal line test<\/li>\r\n\t<li>Find the inverse of a function and graph its reflection<\/li>\r\n\t<li>Evaluate inverse trigonometric functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Inverse Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition of Inverse Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">An inverse function \"undoes\" what the original function does<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation: [latex]f^{-1}[\/latex] is the inverse of [latex]f[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f^{-1}(f(x)) = x[\/latex] and [latex]f(f^{-1}(x)) = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain and Range Relationship:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">One-to-One Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Each element in the codomain is paired with at most one element in the domain<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Only one-to-one functions have inverses that are also functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Horizontal Line Test:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Determines if a function is one-to-one<\/li>\r\n\t<li class=\"whitespace-normal break-words\">A function passes if no horizontal line intersects its graph more than once<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inverse Function Properties:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Reverses the input and output of the original function<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphically, it's a reflection of the original function over [latex]y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572294776\">Is the function [latex]f[\/latex] graphed in the following image one-to-one?<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202541\/CNX_Calc_Figure_01_04_007.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of the function \u201cf(x) = (x cubed) - x\u201d which is a curved function. The function increases, decreases, then increases again. The x intercepts are at the points (-1, 0), (0,0), and (1, 0). The y intercept is at the origin.\" width=\"325\" height=\"366\" \/> Figure 8. Is this function one-to-one?[\/caption]\r\n\r\n<p>[reveal-answer q=\"880346\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"880346\"]<\/p>\r\n<p id=\"fs-id1165042094151\">Use the horizontal line test.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572216730\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572216730\"]<\/p>\r\n<p id=\"fs-id1170572216730\">No.<\/p>\r\n<p><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial;\">[\/hidden-answer]<\/span><\/p>\r\n<\/section>\r\n<h2>Finding a Function\u2019s Inverse<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>The process for finding an inverse function is:<\/p>\r\n<ol class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Ensure the function is one-to-one<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Replace [latex]f(x)[\/latex] with [latex]y[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solve the equation for [latex]x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Interchange [latex]x [\/latex] and [latex]y[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ol>\r\n<p>You can always check and verify if you have found a functions inverse by:<\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Checking that [latex]f^{-1}(f(x)) = x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Also verifying that [latex]f(f^{-1}(x)) = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572478776\">Find the inverse of the function [latex]f(x)=\\dfrac{3x}{(x-2)}[\/latex]. State the domain and range of the inverse function.<\/p>\r\n<p>[reveal-answer q=\"466233\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"466233\"]<\/p>\r\n<p id=\"fs-id1165042050880\">Use the Problem-Solving Strategy above for finding inverse functions.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572479045\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572479045\"]<\/p>\r\n<p id=\"fs-id1170572479045\">[latex]f^{-1}(x)=\\dfrac{2x}{x-3}[\/latex].<\/p>\r\n<p>The domain of [latex]f^{-1}[\/latex] is [latex]\\{x|x \\ne 3\\}[\/latex].<\/p>\r\n<p>The range of [latex]f^{-1}[\/latex] is [latex]\\{y|y \\ne 2\\}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the inverse of [latex]f(x) = (x + 2)\u00b3 - 1[\/latex]. State the domain and range of the inverse function.<\/p>\r\n<p><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;f604f497-fe26-4bdf-be5f-a8a0d29c74d6&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:513,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0}\">f604f497-fe26-4bdf-be5f-a8a0d29c74d6<\/span><\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"394425\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"394425\"]<\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Ensure [latex]f(x)[\/latex] is one-to-one:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is a cubic function with no turning points, so it's one-to-one.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Replace [latex]f(x)[\/latex] with [latex]y[\/latex]:<center>[latex]y = (x + 2)\u00b3 - 1[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:<center>[latex]\\begin{align*}y + 1 &amp;= (x + 2)\u00b3 \\\\ \\sqrt[3]{(y + 1)} &amp;= x + 2 \\\\ \\sqrt[3]{(y + 1)} - 2 &amp;= x\\end{align*}[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Interchange [latex]x[\/latex] and [latex]y[\/latex]:<center>[latex]y = \\sqrt[3]{(x + 1)} - 2[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex]:<center>[latex]f^{-1}(x) = \\sqrt[3]{(x + 1)} - 2[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Verify:<center>[latex]f(f^{-1}(x)) = ((\\sqrt[3]{(x + 1)} - 2) + 2)\u00b3 - 1 = (\\sqrt[3]{(x + 1)})\u00b3 - 1 = (x + 1) - 1 = x[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain and Range:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Domain of [latex]f[\/latex]: [latex](-\u221e, \u221e)[\/latex] \u2192 Range of [latex]f^{-1}[\/latex]: [latex](-\u221e, \u221e)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Range of [latex]f[\/latex]: [latex](-\u221e, \u221e)[\/latex] \u2192 Domain of [latex]f^{-1}[\/latex]: [latex](-\u221e, \u221e)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>\r\n<h2>Graphing Inverse Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Graphical Relationship of Inverses:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Inverse functions are reflections of each other over the line [latex]y = x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Points [latex](a,b)[\/latex] on [latex]f(x)[\/latex] correspond to points [latex](b,a)[\/latex] on [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphing Inverse Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Plot the original function<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Reflect points over [latex]y = x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Connect reflected points to form the inverse function's graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Restricted Domains:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Used when a function is not one-to-one over its entire domain<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Allows creation of an inverse function for a portion of the original function<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Different restrictions can lead to different inverse functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">One-to-One on Restricted Domains:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Ensure the function passes the horizontal line test on the restricted domain<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain of [latex]f[\/latex] becomes range of [latex]f^{-1} [\/latex]and vice versa<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572480271\">Sketch the graph of [latex]f(x)=2x+3[\/latex] and the graph of its inverse using the symmetry property of inverse functions.<\/p>\r\n<p>[reveal-answer q=\"708256\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"708256\"]<\/p>\r\n<p id=\"fs-id1165041823836\">The graphs are symmetric about the line [latex]y=x[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572481068\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572481068\"]<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202551\/CNX_Calc_Figure_01_04_011.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of two functions. The first function is \u201cf(x) = 2x +3\u201d, an increasing straight line function. The function has an x intercept at (-1.5, 0) and a y intercept at (0, 3). The second function is \u201cf inverse (x) = (x - 3)\/2\u201d, an increasing straight line function, which increases at a slower rate than the first function. The function has an x intercept at (3, 0) and a y intercept at (0, -1.5). In addition to the two functions, there is a diagonal dotted line potted with the equation \u201cy =x\u201d, which shows that \u201cf(x)\u201d and \u201cf inverse (x)\u201d are mirror images about the line \u201cy =x\u201d.\" width=\"325\" height=\"358\" \/> Figure 12. Graph of [latex]f(x)=2x+3[\/latex] and its inverse.[\/caption]\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572551773\">Consider [latex]f(x)=\\dfrac{1}{x^2}[\/latex] restricted to the domain [latex](\u2212\\infty ,0)[\/latex]. Verify that [latex]f[\/latex] is one-to-one on this domain. Determine the domain and range of the inverse of [latex]f[\/latex] and find a formula for [latex]f^{-1}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"118844\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"118844\"]<\/p>\r\n<p id=\"fs-id1165042134545\">The domain and range of [latex]f^{-1}[\/latex] is given by the range and domain of [latex]f[\/latex], respectively. To find [latex]f^{-1}[\/latex], solve [latex]y=\\dfrac{1}{x^2}[\/latex] for [latex]x[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572141203\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572141203\"]<\/p>\r\n<p id=\"fs-id1170572141203\">The domain of [latex]f^{-1}[\/latex] is [latex](0,\\infty)[\/latex]. The range of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,0)[\/latex]. The inverse function is given by the formula [latex]f^{-1}(x)=\\dfrac{-1<br \/>\r\n}{\\sqrt{x}}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Inverse Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition of Inverse Trigonometric Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Created by restricting domains of standard trig functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inverse sine ([latex]\\sin^{-1}[\/latex] or arcsin), inverse cosine ([latex]\\cos^{-1}[\/latex] or arccos), inverse tangent ([latex]\\tan^{-1}[\/latex] or arctan), etc.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domains and Ranges:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">arcsin and arccos: Domain [latex][-1, 1],[\/latex] Range [latex][\\frac{-\u03c0}{2}, \\frac{\u03c0}{2}][\/latex] (arcsin) and [latex][0, \u03c0][\/latex] (arccos)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">arctan: Domain [latex](-\\infty, \\infty)[\/latex], Range [latex][\\frac{-\u03c0}{2}, \\frac{\u03c0}{2}][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphs:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Reflections of restricted trigonometric functions over [latex]y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Composition Properties:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sin(\\sin^{-1}(x)) = x[\/latex] for [latex]x[\/latex] in [latex][-1, 1][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sin^{-1}(\\sin(x)) = x[\/latex] for [latex]x[\/latex] in [latex][\\frac{-\u03c0}{2}, \\frac{\u03c0}{2}][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Similar properties for other inverse trig functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Evaluate and simplify the following:<\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\cos^{-1}(-1\/2)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sin(\\tan^{-1}(1))[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sin^{-1}(\\sin(\\frac{5\u03c0}{6}))[\/latex]<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"968249\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"968249\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<p class=\"whitespace-pre-wrap break-words\">Evaluating [latex]\\cos^{-1}(-\\frac{1}{2})[\/latex] is equivalent to finding the angle [latex]\\theta[\/latex] such that [latex]\\cos\\theta = -\\frac{1}{2}[\/latex] and [latex]0 \\leq \\theta \\leq \\pi[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The angle [latex]\\theta = \\frac{2\\pi}{3}[\/latex] satisfies these two conditions. Therefore, [latex]\\cos^{-1}(-\\frac{1}{2}) = \\frac{2\\pi}{3}[\/latex].<\/p>\r\n<\/li>\r\n\t<li>\r\n<p class=\"whitespace-pre-wrap break-words\">First we use the fact that [latex]\\tan^{-1}(1) = \\frac{\\pi}{4}[\/latex]. Then [latex]\\sin(\\frac{\\pi}{4}) = \\frac{1}{\\sqrt{2}}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]\\sin(\\tan^{-1}(1)) = \\frac{1}{\\sqrt{2}}[\/latex].<\/p>\r\n<\/li>\r\n\t<li>\r\n<p class=\"whitespace-pre-wrap break-words\">To evaluate [latex]\\sin^{-1}(\\sin(\\frac{5\\pi}{6}))[\/latex], first use the fact that [latex]\\sin(\\frac{5\\pi}{6}) = \\frac{1}{2}[\/latex]. Then we need to find the angle [latex]\\theta[\/latex] such that [latex]\\sin(\\theta) = \\frac{1}{2}[\/latex] and [latex]-\\frac{\\pi}{2} \\leq \\theta \\leq \\frac{\\pi}{2}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Since [latex]\\frac{\\pi}{6}[\/latex] satisfies both these conditions, we have [latex]\\sin^{-1}(\\sin(\\frac{5\\pi}{6})) = \\sin^{-1}(\\frac{1}{2}) = \\frac{\\pi}{6}[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>&nbsp;<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify conditions for a function&#8217;s inverse using the horizontal line test<\/li>\n<li>Find the inverse of a function and graph its reflection<\/li>\n<li>Evaluate inverse trigonometric functions<\/li>\n<\/ul>\n<\/section>\n<h2>Inverse Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Inverse Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">An inverse function &#8220;undoes&#8221; what the original function does<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]f^{-1}[\/latex] is the inverse of [latex]f[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f^{-1}(f(x)) = x[\/latex] and [latex]f(f^{-1}(x)) = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and Range Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">One-to-One Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each element in the codomain is paired with at most one element in the domain<\/li>\n<li class=\"whitespace-normal break-words\">Only one-to-one functions have inverses that are also functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Line Test:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determines if a function is one-to-one<\/li>\n<li class=\"whitespace-normal break-words\">A function passes if no horizontal line intersects its graph more than once<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inverse Function Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reverses the input and output of the original function<\/li>\n<li class=\"whitespace-normal break-words\">Graphically, it&#8217;s a reflection of the original function over [latex]y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572294776\">Is the function [latex]f[\/latex] graphed in the following image one-to-one?<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202541\/CNX_Calc_Figure_01_04_007.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of the function \u201cf(x) = (x cubed) - x\u201d which is a curved function. The function increases, decreases, then increases again. The x intercepts are at the points (-1, 0), (0,0), and (1, 0). The y intercept is at the origin.\" width=\"325\" height=\"366\" \/><figcaption class=\"wp-caption-text\">Figure 8. Is this function one-to-one?<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q880346\">Hint<\/button><\/p>\n<div id=\"q880346\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042094151\">Use the horizontal line test.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572216730\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572216730\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572216730\">No.<\/p>\n<p><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/section>\n<h2>Finding a Function\u2019s Inverse<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The process for finding an inverse function is:<\/p>\n<ol class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ensure the function is one-to-one<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]f(x)[\/latex] with [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve the equation for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Interchange [latex]x[\/latex] and [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ol>\n<p>You can always check and verify if you have found a functions inverse by:<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Checking that [latex]f^{-1}(f(x)) = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Also verifying that [latex]f(f^{-1}(x)) = x[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572478776\">Find the inverse of the function [latex]f(x)=\\dfrac{3x}{(x-2)}[\/latex]. State the domain and range of the inverse function.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q466233\">Hint<\/button><\/p>\n<div id=\"q466233\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042050880\">Use the Problem-Solving Strategy above for finding inverse functions.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572479045\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572479045\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572479045\">[latex]f^{-1}(x)=\\dfrac{2x}{x-3}[\/latex].<\/p>\n<p>The domain of [latex]f^{-1}[\/latex] is [latex]\\{x|x \\ne 3\\}[\/latex].<\/p>\n<p>The range of [latex]f^{-1}[\/latex] is [latex]\\{y|y \\ne 2\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the inverse of [latex]f(x) = (x + 2)\u00b3 - 1[\/latex]. State the domain and range of the inverse function.<\/p>\n<p><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;f604f497-fe26-4bdf-be5f-a8a0d29c74d6&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:513,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0}\">f604f497-fe26-4bdf-be5f-a8a0d29c74d6<\/span><\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q394425\">Show Answer<\/button><\/p>\n<div id=\"q394425\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Ensure [latex]f(x)[\/latex] is one-to-one:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is a cubic function with no turning points, so it&#8217;s one-to-one.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]f(x)[\/latex] with [latex]y[\/latex]:\n<div style=\"text-align: center;\">[latex]y = (x + 2)\u00b3 - 1[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\n<div style=\"text-align: center;\">[latex]\\begin{align*}y + 1 &= (x + 2)\u00b3 \\\\ \\sqrt[3]{(y + 1)} &= x + 2 \\\\ \\sqrt[3]{(y + 1)} - 2 &= x\\end{align*}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interchange [latex]x[\/latex] and [latex]y[\/latex]:\n<div style=\"text-align: center;\">[latex]y = \\sqrt[3]{(x + 1)} - 2[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex]:\n<div style=\"text-align: center;\">[latex]f^{-1}(x) = \\sqrt[3]{(x + 1)} - 2[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verify:\n<div style=\"text-align: center;\">[latex]f(f^{-1}(x)) = ((\\sqrt[3]{(x + 1)} - 2) + 2)\u00b3 - 1 = (\\sqrt[3]{(x + 1)})\u00b3 - 1 = (x + 1) - 1 = x[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of [latex]f[\/latex]: [latex](-\u221e, \u221e)[\/latex] \u2192 Range of [latex]f^{-1}[\/latex]: [latex](-\u221e, \u221e)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range of [latex]f[\/latex]: [latex](-\u221e, \u221e)[\/latex] \u2192 Domain of [latex]f^{-1}[\/latex]: [latex](-\u221e, \u221e)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n<h2>Graphing Inverse Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Graphical Relationship of Inverses:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Inverse functions are reflections of each other over the line [latex]y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Points [latex](a,b)[\/latex] on [latex]f(x)[\/latex] correspond to points [latex](b,a)[\/latex] on [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphing Inverse Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Plot the original function<\/li>\n<li class=\"whitespace-normal break-words\">Reflect points over [latex]y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Connect reflected points to form the inverse function&#8217;s graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Restricted Domains:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when a function is not one-to-one over its entire domain<\/li>\n<li class=\"whitespace-normal break-words\">Allows creation of an inverse function for a portion of the original function<\/li>\n<li class=\"whitespace-normal break-words\">Different restrictions can lead to different inverse functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">One-to-One on Restricted Domains:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ensure the function passes the horizontal line test on the restricted domain<\/li>\n<li class=\"whitespace-normal break-words\">Domain of [latex]f[\/latex] becomes range of [latex]f^{-1}[\/latex]and vice versa<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572480271\">Sketch the graph of [latex]f(x)=2x+3[\/latex] and the graph of its inverse using the symmetry property of inverse functions.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q708256\">Hint<\/button><\/p>\n<div id=\"q708256\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165041823836\">The graphs are symmetric about the line [latex]y=x[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572481068\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572481068\" class=\"hidden-answer\" style=\"display: none\">\n<figure style=\"width: 325px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202551\/CNX_Calc_Figure_01_04_011.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of two functions. The first function is \u201cf(x) = 2x +3\u201d, an increasing straight line function. The function has an x intercept at (-1.5, 0) and a y intercept at (0, 3). The second function is \u201cf inverse (x) = (x - 3)\/2\u201d, an increasing straight line function, which increases at a slower rate than the first function. The function has an x intercept at (3, 0) and a y intercept at (0, -1.5). In addition to the two functions, there is a diagonal dotted line potted with the equation \u201cy =x\u201d, which shows that \u201cf(x)\u201d and \u201cf inverse (x)\u201d are mirror images about the line \u201cy =x\u201d.\" width=\"325\" height=\"358\" \/><figcaption class=\"wp-caption-text\">Figure 12. Graph of [latex]f(x)=2x+3[\/latex] and its inverse.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572551773\">Consider [latex]f(x)=\\dfrac{1}{x^2}[\/latex] restricted to the domain [latex](\u2212\\infty ,0)[\/latex]. Verify that [latex]f[\/latex] is one-to-one on this domain. Determine the domain and range of the inverse of [latex]f[\/latex] and find a formula for [latex]f^{-1}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q118844\">Hint<\/button><\/p>\n<div id=\"q118844\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042134545\">The domain and range of [latex]f^{-1}[\/latex] is given by the range and domain of [latex]f[\/latex], respectively. To find [latex]f^{-1}[\/latex], solve [latex]y=\\dfrac{1}{x^2}[\/latex] for [latex]x[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572141203\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572141203\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572141203\">The domain of [latex]f^{-1}[\/latex] is [latex](0,\\infty)[\/latex]. The range of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,0)[\/latex]. The inverse function is given by the formula [latex]f^{-1}(x)=\\dfrac{-1<br \/>  }{\\sqrt{x}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Inverse Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Inverse Trigonometric Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Created by restricting domains of standard trig functions<\/li>\n<li class=\"whitespace-normal break-words\">Inverse sine ([latex]\\sin^{-1}[\/latex] or arcsin), inverse cosine ([latex]\\cos^{-1}[\/latex] or arccos), inverse tangent ([latex]\\tan^{-1}[\/latex] or arctan), etc.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domains and Ranges:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">arcsin and arccos: Domain [latex][-1, 1],[\/latex] Range [latex][\\frac{-\u03c0}{2}, \\frac{\u03c0}{2}][\/latex] (arcsin) and [latex][0, \u03c0][\/latex] (arccos)<\/li>\n<li class=\"whitespace-normal break-words\">arctan: Domain [latex](-\\infty, \\infty)[\/latex], Range [latex][\\frac{-\u03c0}{2}, \\frac{\u03c0}{2}][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reflections of restricted trigonometric functions over [latex]y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Composition Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sin(\\sin^{-1}(x)) = x[\/latex] for [latex]x[\/latex] in [latex][-1, 1][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sin^{-1}(\\sin(x)) = x[\/latex] for [latex]x[\/latex] in [latex][\\frac{-\u03c0}{2}, \\frac{\u03c0}{2}][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Similar properties for other inverse trig functions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Evaluate and simplify the following:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\cos^{-1}(-1\/2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sin(\\tan^{-1}(1))[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sin^{-1}(\\sin(\\frac{5\u03c0}{6}))[\/latex]<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q968249\">Show Answer<\/button><\/p>\n<div id=\"q968249\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<p class=\"whitespace-pre-wrap break-words\">Evaluating [latex]\\cos^{-1}(-\\frac{1}{2})[\/latex] is equivalent to finding the angle [latex]\\theta[\/latex] such that [latex]\\cos\\theta = -\\frac{1}{2}[\/latex] and [latex]0 \\leq \\theta \\leq \\pi[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The angle [latex]\\theta = \\frac{2\\pi}{3}[\/latex] satisfies these two conditions. Therefore, [latex]\\cos^{-1}(-\\frac{1}{2}) = \\frac{2\\pi}{3}[\/latex].<\/p>\n<\/li>\n<li>\n<p class=\"whitespace-pre-wrap break-words\">First we use the fact that [latex]\\tan^{-1}(1) = \\frac{\\pi}{4}[\/latex]. Then [latex]\\sin(\\frac{\\pi}{4}) = \\frac{1}{\\sqrt{2}}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]\\sin(\\tan^{-1}(1)) = \\frac{1}{\\sqrt{2}}[\/latex].<\/p>\n<\/li>\n<li>\n<p class=\"whitespace-pre-wrap break-words\">To evaluate [latex]\\sin^{-1}(\\sin(\\frac{5\\pi}{6}))[\/latex], first use the fact that [latex]\\sin(\\frac{5\\pi}{6}) = \\frac{1}{2}[\/latex]. Then we need to find the angle [latex]\\theta[\/latex] such that [latex]\\sin(\\theta) = \\frac{1}{2}[\/latex] and [latex]-\\frac{\\pi}{2} \\leq \\theta \\leq \\frac{\\pi}{2}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Since [latex]\\frac{\\pi}{6}[\/latex] satisfies both these conditions, we have [latex]\\sin^{-1}(\\sin(\\frac{5\\pi}{6})) = \\sin^{-1}(\\frac{1}{2}) = \\frac{\\pi}{6}[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"fresh_take","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1131"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1131\/revisions"}],"predecessor-version":[{"id":4302,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1131\/revisions\/4302"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1131\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1131"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1131"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1131"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}