{"id":1559,"date":"2018-01-11T20:26:43","date_gmt":"2018-01-11T20:26:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/inverse-functions\/"},"modified":"2018-02-07T16:37:54","modified_gmt":"2018-02-07T16:37:54","slug":"inverse-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/inverse-functions\/","title":{"raw":"1.4 Inverse Functions","rendered":"1.4 Inverse Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Determine the conditions for when a function has an inverse.<\/li>\r\n \t<li>Use the horizontal line test to recognize when a function is one-to-one.<\/li>\r\n \t<li>Find the inverse of a given function.<\/li>\r\n \t<li>Draw the graph of an inverse function.<\/li>\r\n \t<li>Evaluate inverse trigonometric functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170572215818\">An <strong>inverse function<\/strong> reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. In this section, we define an inverse function formally and state the necessary conditions for an inverse function to exist. We examine how to find an inverse function and study the relationship between the graph of a function and the graph of its inverse. Then we apply these ideas to define and discuss properties of the inverse trigonometric functions.<\/p>\r\n\r\n<div id=\"fs-id1170572245927\" class=\"bc-section section\">\r\n<h1>Existence of an Inverse Function<\/h1>\r\nWe begin with an example. Given a function [latex]f[\/latex] and an output [latex]y=f(x)[\/latex], we are often interested in finding what value or values [latex]x[\/latex] were mapped to [latex]y[\/latex] by [latex]f[\/latex]. For example, consider the function [latex]f(x)=x^3+4[\/latex]. Since any output [latex]y=x^3+4[\/latex], we can solve this equation for [latex]x[\/latex] to find that the input is [latex]x=\\sqrt[3]{y-4}[\/latex]. This equation defines [latex]x[\/latex] as a function of [latex]y[\/latex]. Denoting this function as [latex]f^{-1}[\/latex], and writing [latex]x=f^{-1}(y)=\\sqrt[3]{y-4}[\/latex], we see that for any [latex]x[\/latex] in the domain of [latex]f, \\, f^{-1}(f(x))=f^{-1}(x^3+4)=x[\/latex]. Thus, this new function, [latex]f^{-1}[\/latex], \u201cundid\u201d what the original function [latex]f[\/latex] did. A function with this property is called the inverse function of the original function.\r\n<div id=\"fs-id1170572137845\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1170572110489\">Given a function [latex]f[\/latex] with domain [latex]D[\/latex] and range [latex]R[\/latex], its inverse function (if it exists) is the function [latex]f^{-1}[\/latex] with domain [latex]R[\/latex] and range [latex]D[\/latex] such that [latex]f^{-1}(y)=x[\/latex] if [latex]f(x)=y[\/latex]. In other words, for a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex],<\/p>\r\n\r\n<div id=\"fs-id1170572141883\" class=\"equation\">[latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in [latex]D[\/latex], and [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in [latex]R[\/latex].<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572548323\">Note that [latex]f^{-1}[\/latex] is read as \u201cf inverse.\u201d Here, the -1 is not used as an exponent and [latex]f^{-1}(x) \\ne 1\/f(x)[\/latex].\u00a0<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_001\">(Figure)<\/a> shows the relationship between the domain and range of [latex]f[\/latex] and the domain and range of [latex]f^{-1}[\/latex].<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_04_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202523\/CNX_Calc_Figure_01_04_001.jpg\" alt=\"An image of two bubbles. The first bubble is orange and has two labels: the top label is \u201cDomain of f\u201d and the bottom label is \u201cRange of f inverse\u201d. Within this bubble is the variable \u201cx\u201d. An orange arrow with the label \u201cf\u201d points from this bubble to the second bubble. The second bubble is blue and has two labels: the top label is \u201crange of f\u201d and the bottom label is \u201cdomain of f inverse\u201d. Within this bubble is the variable \u201cy\u201d. A blue arrow with the label \u201cf inverse\u201d points from this bubble to the first bubble.\" width=\"487\" height=\"160\" \/> <strong>Figure 1.<\/strong> Given a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f^{-1}(y)=x[\/latex] if and only if [latex]f(x)=y[\/latex]. The range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex] and the domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex].[\/caption]<\/div>\r\n<p id=\"fs-id1170572141586\">Recall that a function has exactly one output for each input. Therefore, to define an inverse function, we need to map each input to exactly one output. For example, let\u2019s try to find the inverse function for [latex]f(x)=x^2[\/latex]. Solving the equation [latex]y=x^2[\/latex] for [latex]x[\/latex], we arrive at the equation [latex]x= \\pm \\sqrt{y}[\/latex]. This equation does not describe [latex]x[\/latex] as a function of [latex]y[\/latex] because there are two solutions to this equation for every [latex]y&gt;0[\/latex]. The problem with trying to find an inverse function for [latex]f(x)=x^2[\/latex] is that two inputs are sent to the same output for each output [latex]y&gt;0[\/latex]. The function [latex]f(x)=x^3+4[\/latex] discussed earlier did not have this problem. For that function, each input was sent to a different output. A function that sends each input to a <em>different<\/em> output is called a<strong> one-to-one function.<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572110310\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1170572229709\">We say a [latex]f[\/latex] is a one-to-one function if [latex]f(x_1) \\ne f(x_2)[\/latex] when [latex]x_1 \\ne x_2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572103424\">One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the [latex]xy[\/latex]-plane, according to the <strong>horizontal line test<\/strong>, it cannot intersect the graph more than once. We note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_002\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"fs-id1170572226107\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Horizontal Line Test<\/h3>\r\n<p id=\"fs-id1170572137262\">A function [latex]f[\/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[\/latex] no more than once.<\/p>\r\n\r\n<\/div>\r\n<div id=\"CNX_Calc_Figure_01_04_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202527\/CNX_Calc_Figure_01_04_002.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -3 to 4. The first graph is of the function \u201cf(x) = x squared\u201d, which is a parabola. The function decreases until it hits the origin, where it begins to increase. The x intercept and y intercept are both at the origin. There are two orange horizontal lines also plotted on the graph, both of which run through the function at two points each. The second graph is of the function \u201cf(x) = x cubed\u201d, which is an increasing curved function. The x intercept and y intercept are both at the origin. There are three orange lines also plotted on the graph, each of which only intersects the function at one point.\" width=\"487\" height=\"313\" \/> <strong>Figure 2.<\/strong> (a) The function [latex]f(x)=x^2[\/latex] is not one-to-one because it fails the horizontal line test. (b) The function [latex]f(x)=x^3[\/latex] is one-to-one because it passes the horizontal line test.[\/caption]<\/div>\r\n<div id=\"fs-id1170572228079\" class=\"textbox examples\">\r\n<h3>Determining Whether a Function Is One-to-One<\/h3>\r\n<div id=\"fs-id1170572102492\" class=\"exercise\">\r\n<div id=\"fs-id1170572103254\" class=\"textbox\">\r\n<p id=\"fs-id1170572150756\">For each of the following functions, use the horizontal line test to determine whether it is one-to-one.<\/p>\r\n\r\n<ol id=\"fs-id1170572174892\" style=\"list-style-type: lower-alpha\">\r\n \t<li><span id=\"fs-id1170572108141\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202531\/CNX_Calc_Figure_01_04_003.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9)\" \/><\/span><\/li>\r\n \t<li><span id=\"fs-id1170572140980\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202534\/CNX_Calc_Figure_01_04_004.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote.\" \/><\/span><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572241370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572241370\"]\r\n<ol id=\"fs-id1170572241370\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Since the horizontal line [latex]y=n[\/latex] for any integer [latex]n\\ge 0[\/latex] intersects the graph more than once, this function is not one-to-one.\r\n<span id=\"fs-id1170572107320\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202536\/CNX_Calc_Figure_01_04_005.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9). There are also two horizontal orange lines plotted on the graph, each of which run through an entire step of the function.\" \/><\/span><\/li>\r\n \t<li>Since every horizontal line intersects the graph once (at most), this function is one-to-one.\r\n<span id=\"fs-id1170572454307\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202538\/CNX_Calc_Figure_01_04_006.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote. There are also three horizontal orange lines plotted on the graph, each of which only runs through the function at one point.\" \/><\/span><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572151745\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572141532\" class=\"exercise\">\r\n<div id=\"fs-id1170572479279\" class=\"textbox\">\r\n<p id=\"fs-id1170572294776\">Is the function [latex]f[\/latex] graphed in the following image one-to-one?<\/p>\r\n<span id=\"fs-id1170572224899\"><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.\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572216730\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572216730\"]\r\n<p id=\"fs-id1170572216730\">No.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042132352\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042094151\">Use the horizontal line test.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572177789\" class=\"bc-section section\">\r\n<h1>Finding a Function\u2019s Inverse<\/h1>\r\n<p id=\"fs-id1170572150503\">We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of [latex]f[\/latex] to elements in the range of [latex]f[\/latex]. The inverse function maps each element from the range of [latex]f[\/latex] back to its corresponding element from the domain of [latex]f[\/latex]. Therefore, to find the inverse function of a one-to-one function [latex]f[\/latex], given any [latex]y[\/latex] in the range of [latex]f[\/latex], we need to determine which [latex]x[\/latex] in the domain of [latex]f[\/latex] satisfies [latex]f(x)=y[\/latex]. Since [latex]f[\/latex] is one-to-one, there is exactly one such value [latex]x[\/latex]. We can find that value [latex]x[\/latex] by solving the equation [latex]f(x)=y[\/latex] for [latex]x[\/latex]. Doing so, we are able to write [latex]x[\/latex] as a function of [latex]y[\/latex] where the domain of this function is the range of [latex]f[\/latex] and the range of this new function is the domain of [latex]f[\/latex]. Consequently, this function is the inverse of [latex]f[\/latex], and we write [latex]x=f^{-1}(y)[\/latex]. Since we typically use the variable [latex]x[\/latex] to denote the independent variable and [latex]y[\/latex] to denote the dependent variable, we often interchange the roles of [latex]x[\/latex] and [latex]y[\/latex], and write [latex]y=f^{-1}(x)[\/latex]. Representing the inverse function in this way is also helpful later when we graph a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] on the same axes.<\/p>\r\n\r\n<div id=\"fs-id1170572552427\" class=\"textbox key-takeaways problem-solving\">\r\n<h3>Problem-Solving Strategy: Finding an Inverse Function<\/h3>\r\n<ol id=\"fs-id1170572450945\">\r\n \t<li>Solve the equation [latex]y=f(x)[\/latex] for [latex]x[\/latex].<\/li>\r\n \t<li>Interchange the variables [latex]x[\/latex] and [latex]y[\/latex] and write [latex]y=f^{-1}(x)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572208522\" class=\"textbox examples\">\r\n<h3>Finding an Inverse Function<\/h3>\r\n<div id=\"fs-id1170572240528\" class=\"exercise\">\r\n<div id=\"fs-id1170572546518\" class=\"textbox\">\r\n<p id=\"fs-id1170572546520\">Find the inverse for the function [latex]f(x)=3x-4[\/latex]. State the domain and range of the inverse function. Verify that [latex]f^{-1}(f(x))=x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572481500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572481500\"]\r\n<p id=\"fs-id1170572481500\">Follow the steps outlined in the strategy.<\/p>\r\n<p id=\"fs-id1170572548820\">Step 1. If [latex]y=3x-4[\/latex], then [latex]3x=y+4[\/latex] and [latex]x=\\frac{1}{3}y+\\frac{4}{3}[\/latex].<\/p>\r\n<p id=\"fs-id1170572479776\">Step 2. Rewrite as [latex]y=\\frac{1}{3}x+\\frac{4}{3}[\/latex] and let [latex]y=f^{-1}(x)[\/latex].<\/p>\r\n<p id=\"fs-id1170572453043\">Therefore, [latex]f^{-1}(x)=\\frac{1}{3}x+\\frac{4}{3}[\/latex].<\/p>\r\n<p id=\"fs-id1170572240611\">Since the domain of [latex]f[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex], the range of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex]. Since the range of [latex]f[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex], the domain of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex].<\/p>\r\n<p id=\"fs-id1170572222969\">You can verify that [latex]f^{-1}(f(x))=x[\/latex] by writing<\/p>\r\n\r\n<div id=\"fs-id1170572453591\" class=\"equation unnumbered\">[latex]f^{-1}(f(x))=f^{-1}(3x-4)=\\frac{1}{3}(3x-4)+\\frac{4}{3}=x-\\frac{4}{3}+\\frac{4}{3}=x[\/latex].<\/div>\r\n<p id=\"fs-id1170572546429\">Note that for [latex]f^{-1}(x)[\/latex] to be the inverse of [latex]f(x)[\/latex], both [latex]f^{-1}(f(x))=x[\/latex] and [latex]f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of the inside function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572478768\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572478772\" class=\"exercise\">\r\n<div id=\"fs-id1170572478774\" class=\"textbox\">\r\n<p id=\"fs-id1170572478776\">Find the inverse of the function [latex]f(x)=3x\/(x-2)[\/latex]. State the domain and range of the inverse function.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572479045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572479045\"]\r\n<p id=\"fs-id1170572479045\">[latex]f^{-1}(x)=\\frac{2x}{x-3}[\/latex]. The domain of [latex]f^{-1}[\/latex] is [latex]\\{x|x \\ne 3\\}[\/latex]. The range of [latex]f^{-1}[\/latex] is [latex]\\{y|y \\ne 2\\}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042047502\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042050880\">Use the <a class=\"autogenerated-content\" href=\"#fs-id1170572552427\">(Note)<\/a> for finding inverse functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572363370\" class=\"bc-section section\">\r\n<h2>Graphing Inverse Functions<\/h2>\r\n<p id=\"fs-id1170572363375\">Let\u2019s consider the relationship between the graph of a function [latex]f[\/latex] and the graph of its inverse. Consider the graph of [latex]f[\/latex] shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_008\">(Figure)<\/a> and a point [latex](a,b)[\/latex] on the graph. Since [latex]b=f(a)[\/latex], then [latex]f^{-1}(b)=a[\/latex]. Therefore, when we graph [latex]f^{-1}[\/latex], the point [latex](b,a)[\/latex] is on the graph. As a result, the graph of [latex]f^{-1}[\/latex] is a reflection of the graph of [latex]f[\/latex] about the line [latex]y=x[\/latex].<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_04_008\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"710\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202543\/CNX_Calc_Figure_01_04_017.jpg\" alt=\"An image of two graphs. The first graph is of \u201cy = f(x)\u201d, which is a curved increasing function, that increases at a faster rate as x increases. The point (a, b) is on the graph of the function in the first quadrant. The second graph also graphs \u201cy = f(x)\u201d with the point (a, b), but also graphs the function \u201cy = f inverse (x)\u201d, an increasing curved function, that increases at a slower rate as x increases. This function includes the point (b, a). 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=\"710\" height=\"387\" \/> <strong>Figure 3.<\/strong> (a) The graph of this function [latex]f[\/latex] shows point [latex](a,b)[\/latex] on the graph of [latex]f[\/latex]. (b) Since [latex](a,b)[\/latex] is on the graph of [latex]f[\/latex], the point [latex](b,a)[\/latex] is on the graph of [latex]f^{-1}[\/latex]. The graph of [latex]f^{-1}[\/latex] is a reflection of the graph of [latex]f[\/latex] about the line [latex]y=x[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572213184\" class=\"textbox examples\">\r\n<h3>Sketching Graphs of Inverse Functions<\/h3>\r\n<div id=\"fs-id1170572213190\" class=\"exercise\">\r\n<div id=\"fs-id1170572213192\" class=\"textbox\">\r\n<p id=\"fs-id1170572213194\">For the graph of [latex]f[\/latex] in the following image, sketch a graph of [latex]f^{-1}[\/latex] by sketching the line [latex]y=x[\/latex] and using symmetry. Identify the domain and range of [latex]f^{-1}[\/latex].<\/p>\r\n<span id=\"fs-id1170572222895\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202545\/CNX_Calc_Figure_01_04_009.jpg\" alt=\"An image of a graph. The x axis runs from -2 to 2 and the y axis runs from 0 to 2. The graph is of the function \u201cf(x) = square root of (x +2)\u201d, an increasing curved function. The function starts at the point (-2, 0). The x intercept is at (-2, 0) and the y intercept is at the approximate point (0, 1.4).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572222910\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572222910\"]\r\n<p id=\"fs-id1170572222910\">Reflect the graph about the line [latex]y=x[\/latex]. The domain of [latex]f^{-1}[\/latex] is [latex][0,\\infty)[\/latex]. The range of [latex]f^{-1}[\/latex] is [latex][-2,\\infty)[\/latex]. By using the preceding strategy for finding inverse functions, we can verify that the inverse function is [latex]f^{-1}(x)=x^2-2[\/latex], as shown in the graph.<\/p>\r\n<span id=\"fs-id1170572480252\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202548\/CNX_Calc_Figure_01_04_010.jpg\" alt=\"An image of a graph. The x axis runs from -2 to 2 and the y axis runs from -2 to 2. The graph is of two functions. The first function is \u201cf(x) = square root of (x +2)\u201d, an increasing curved function. The function starts at the point (-2, 0). The x intercept is at (-2, 0) and the y intercept is at the approximate point (0, 1.4). The second function is \u201cf inverse (x) = (x squared) -2\u201d, an increasing curved function that starts at the point (0, -2). The x intercept is at the approximate point (1.4, 0) and the y intercept is at the point (0, -2). 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.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480264\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572480267\" class=\"exercise\">\r\n<div id=\"fs-id1170572480269\" class=\"textbox\">\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\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572481068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572481068\"]<span id=\"fs-id1170572481077\"><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.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165041797872\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165041823836\">The graphs are symmetric about the line [latex]y=x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572481090\" class=\"bc-section section\">\r\n<h2>Restricting Domains<\/h2>\r\n<p id=\"fs-id1170572481095\">As we have seen, [latex]f(x)=x^2[\/latex] does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of [latex]f[\/latex] such that the function is one-to-one. This subset is called a restricted domain. By restricting the domain of [latex]f[\/latex], we can define a new function [latex]g[\/latex] such that the domain of [latex]g[\/latex] is the restricted domain of [latex]f[\/latex] and [latex]g(x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]g[\/latex]. Then we can define an inverse function for [latex]g[\/latex] on that domain. For example, since [latex]f(x)=x^2[\/latex] is one-to-one on the interval [latex][0,\\infty)[\/latex], we can define a new function [latex]g[\/latex] such that the domain of [latex]g[\/latex] is [latex][0,\\infty)[\/latex] and [latex]g(x)=x^2[\/latex] for all [latex]x[\/latex] in its domain. Since [latex]g[\/latex] is a one-to-one function, it has an inverse function, given by the formula [latex]g^{-1}(x)=\\sqrt{x}[\/latex]. On the other hand, the function [latex]f(x)=x^2[\/latex] is also one-to-one on the domain [latex](\u2212\\infty,0][\/latex]. Therefore, we could also define a new function [latex]h[\/latex] such that the domain of [latex]h[\/latex] is [latex](\u2212\\infty,0][\/latex] and [latex]h(x)=x^2[\/latex] for all [latex]x[\/latex] in the domain of [latex]h[\/latex]. Then [latex]h[\/latex] is a one-to-one function and must also have an inverse. Its inverse is given by the formula [latex]h^{-1}(x)=\u2212\\sqrt{x}[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_012\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_04_012\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"656\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202554\/CNX_Calc_Figure_01_04_012.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -2 to 5 and a y axis that runs from -2 to 5. The first graph is of two functions. The first function is \u201cg(x) = x squared\u201d, an increasing curved function that starts at the point (0, 0). This function increases at a faster rate for larger values of x. The second function is \u201cg inverse (x) = square root of x\u201d, an increasing curved function that starts at the point (0, 0). This function increases at a slower rate for larger values of x. The first function is \u201ch(x) = x squared\u201d, a decreasing curved function that ends at the point (0, 0). This function decreases at a slower rate for larger values of x. The second function is \u201ch inverse (x) = -(square root of x)\u201d, an increasing curved function that starts at the point (0, 0). This function decreases at a slower rate for larger values of x. 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=\"656\" height=\"353\" \/> <strong>Figure 4.<\/strong> (a) For [latex]g(x)=x^2[\/latex] restricted to [latex][0,\\infty), \\, g^{-1}(x)=\\sqrt{x}[\/latex]. (b) For [latex]h(x)=x^2[\/latex] restricted to [latex](\u2212\\infty,0], \\, h^{-1}(x)=\u2212\\sqrt{x}[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572551599\" class=\"textbox examples\">\r\n<h3>Restricting the Domain<\/h3>\r\n<div id=\"fs-id1170572551604\" class=\"exercise\">\r\n<div id=\"fs-id1170572551606\" class=\"textbox\">\r\n<p id=\"fs-id1170572551608\">Consider the function [latex]f(x)=(x+1)^2[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572551645\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Sketch the graph of [latex]f[\/latex] and use the horizontal line test to show that [latex]f[\/latex] is not one-to-one.<\/li>\r\n \t<li>Show that [latex]f[\/latex] is one-to-one on the restricted domain [latex][-1,\\infty)[\/latex]. Determine the domain and range for the inverse of [latex]f[\/latex] on this restricted domain and find a formula for [latex]f^{-1}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572477860\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572477860\"]\r\n<ol id=\"fs-id1170572477860\" style=\"list-style-type: lower-alpha\">\r\n \t<li>The graph of [latex]f[\/latex] is the graph of [latex]y=x^2[\/latex] shifted left 1 unit. Since there exists a horizontal line intersecting the graph more than once, [latex]f[\/latex] is not one-to-one.\r\n<span id=\"fs-id1170572477898\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202557\/CNX_Calc_Figure_01_04_013.jpg\" alt=\"An image of a graph. The x axis runs from -6 to 6 and the y axis runs from -2 to 10. The graph is of the function \u201cf(x) = (x+ 1) squared\u201d, which is a parabola. The function decreases until the point (-1, 0), where it begins it increases. The x intercept is at the point (-1, 0) and the y intercept is at the point (0, 1). There is also a horizontal dotted line plotted on the graph, which crosses through the function at two points.\" \/><\/span><\/li>\r\n \t<li>On the interval [latex][-1,\\infty), \\, f[\/latex] is one-to-one.\r\n<span id=\"fs-id1170572236096\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202559\/CNX_Calc_Figure_01_04_014.jpg\" alt=\"An image of a graph. The x axis runs from -6 to 6 and the y axis runs from -2 to 10. The graph is of the function \u201cf(x) = (x+ 1) squared\u201d, on the interval [1, infinity). The function starts from the point (-1, 0) and increases. The x intercept is at the point (-1, 0) and the y intercept is at the point (0, 1).\" \/><\/span>\r\nThe domain and range of [latex]f^{-1}[\/latex] are given by the range and domain of [latex]f[\/latex], respectively. Therefore, the domain of [latex]f^{-1}[\/latex] is [latex][0,\\infty)[\/latex] and the range of [latex]f^{-1}[\/latex] is [latex][-1,\\infty)[\/latex]. To find a formula for [latex]f^{-1}[\/latex], solve the equation [latex]y=(x+1)^2[\/latex] for [latex]x[\/latex]. If [latex]y=(x+1)^2[\/latex], then [latex]x=-1 \\pm \\sqrt{y}[\/latex]. Since we are restricting the domain to the interval where [latex]x \\ge -1[\/latex], we need [latex]\\pm \\sqrt{y} \\ge 0[\/latex]. Therefore, [latex]x=-1+\\sqrt{y}[\/latex]. Interchanging [latex]x[\/latex] and [latex]y[\/latex], we write [latex]y=-1+\\sqrt{x}[\/latex] and conclude that [latex]f^{-1}(x)=-1+\\sqrt{x}[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572551765\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572551769\" class=\"exercise\">\r\n<div id=\"fs-id1170572551771\" class=\"textbox\">\r\n<p id=\"fs-id1170572551773\">Consider [latex]f(x)=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\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572141203\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572141203\"]\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)=-1\/\\sqrt{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165041816987\" class=\"commentary\">\r\n<h4>Hint<\/h4>\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=1\/x^2[\/latex] for [latex]x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572141300\" class=\"bc-section section\">\r\n<h1>Inverse Trigonometric Functions<\/h1>\r\n<p id=\"fs-id1170572478796\">The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function (<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/trigonometric-functions\/#CNX_Calc_Figure_01_03_009\">(Figure)<\/a>). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex]. By doing so, we define the inverse sine function on the domain [latex][-1,1][\/latex] such that for any [latex]x[\/latex] in the interval [latex][-1,1][\/latex], the inverse sine function tells us which angle [latex]\\theta [\/latex] in the interval [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex] satisfies [latex] \\sin \\theta =x[\/latex]. Similarly, we can restrict the domains of the other trigonometric functions to define i<strong>nverse trigonometric functions<\/strong>, which are functions that tell us which angle in a certain interval has a specified trigonometric value.<\/p>\r\n\r\n<div id=\"fs-id1170572547395\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1170572547399\">The inverse sine function, denoted [latex] \\sin^{-1}[\/latex] or arcsin, and the inverse cosine function, denoted [latex]\\cos^{-1}[\/latex] or arccos, are defined on the domain [latex]D=\\{x|-1 \\le x \\le 1\\}[\/latex] as follows:<\/p>\r\n\r\n<div id=\"fs-id1170572547453\" class=\"equation\">[latex]\\begin{array}{c}\\sin^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\sin (y)=x \\, \\text{and} \\, -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2};\\hfill \\\\ \\cos^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\cos (y)=x \\, \\text{and} \\, 0 \\le y \\le \\pi \\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572550974\">The inverse tangent function, denoted [latex]\\tan^{-1}[\/latex] or arctan, and inverse cotangent function, denoted [latex]\\cot^{-1}[\/latex] or arccot, are defined on the domain [latex]D=\\{x|-\\infty &lt;x&lt;\\infty \\}[\/latex] as follows:<\/p>\r\n\r\n<div id=\"fs-id1170572551028\" class=\"equation\">[latex]\\begin{array}{c}\\tan^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\tan (y)=x \\, \\text{and} \\, -\\frac{\\pi}{2}&lt;y&lt;\\frac{\\pi}{2};\\hfill \\\\ \\cot^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\cot (y)=x \\, \\text{and} \\, 0&lt;y&lt;\\pi \\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572434001\">The inverse cosecant function, denoted [latex]\\csc^{-1}[\/latex] or arccsc, and inverse secant function, denoted [latex]\\sec^{-1}[\/latex] or arcsec, are defined on the domain [latex]D=\\{x| \\, |x| \\ge 1\\}[\/latex] as follows:<\/p>\r\n\r\n<div id=\"fs-id1170572548730\" class=\"equation\">[latex]\\begin{array}{c}\\csc^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\csc (y)=x \\, \\text{and} \\, -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2}, \\, y\\ne 0;\\hfill \\\\ \\sec^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\sec (y)=x \\, \\text{and} \\, 0 \\le y \\le \\pi, \\, y \\ne \\pi\/2\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572551163\">To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the domains defined earlier and reflect the graphs about the line [latex]y=x[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_015\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_04_015\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"851\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202603\/CNX_Calc_Figure_01_04_018.jpg\" alt=\"An image of six graphs. The first graph is of the function \u201cf(x) = sin inverse(x)\u201d, which is an increasing curve function. The function starts at the point (-1, -(pi\/2)) and increases until it ends at the point (1, (pi\/2)). The x intercept and y intercept are at the origin. The second graph is of the function \u201cf(x) = cos inverse (x)\u201d, which is a decreasing curved function. The function starts at the point (-1, pi) and decreases until it ends at the point (1, 0). The x intercept is at the point (1, 0). The y intercept is at the point (0, (pi\/2)). The third graph is of the function f(x) = tan inverse (x)\u201d, which is an increasing curve function. The function starts close to the horizontal line \u201cy = -(pi\/2)\u201d and increases until it comes close the \u201cy = (pi\/2)\u201d. The function never intersects either of these lines, it always stays between them - they are horizontal asymptotes. The x intercept and y intercept are both at the origin. The fourth graph is of the function \u201cf(x) = cot inverse (x)\u201d, which is a decreasing curved function. The function starts slightly below the horizontal line \u201cy = pi\u201d and decreases until it gets close the x axis. The function never intersects either of these lines, it always stays between them - they are horizontal asymptotes. The fifth graph is of the function \u201cf(x) = csc inverse (x)\u201d, a decreasing curved function. The function starts slightly below the x axis, then decreases until it hits a closed circle point at (-1, -(pi\/2)). The function then picks up again at the point (1, (pi\/2)), where is begins to decrease and approach the x axis, without ever touching the x axis. There is a horizontal asymptote at the x axis. The sixth graph is of the function \u201cf(x) = sec inverse (x)\u201d, an increasing curved function. The function starts slightly above the horizontal line \u201cy = (pi\/2)\u201d, then increases until it hits a closed circle point at (-1, pi). The function then picks up again at the point (1, 0), where is begins to increase and approach the horizontal line \u201cy = (pi\/2)\u201d, without ever touching the line. There is a horizontal asymptote at the \u201cy = (pi\/2)\u201d.\" width=\"851\" height=\"714\" \/> <strong>Figure 5.<\/strong> The graph of each of the inverse trigonometric functions is a reflection about the line [latex]y=x[\/latex] of the corresponding restricted trigonometric function.[\/caption]<\/div>\r\n<div id=\"fs-id1170572366188\" class=\"textbox tryit media-2\">\r\n<p id=\"fs-id1170572366192\">Go to the <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_inversefun\">following site<\/a> for more comparisons of functions and their inverses.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572366201\">When evaluating an inverse trigonometric function, the output is an angle. For example, to evaluate [latex]\\cos^{-1}(\\frac{1}{2})[\/latex], we need to find an angle [latex]\\theta [\/latex] such that [latex] \\cos \\theta =\\frac{1}{2}[\/latex]. Clearly, many angles have this property. However, given the definition of [latex]\\cos^{-1}[\/latex], we need the angle [latex]\\theta [\/latex] that not only solves this equation, but also lies in the interval [latex][0,\\pi][\/latex]. We conclude that [latex]\\cos^{-1}(\\frac{1}{2})=\\frac{\\pi}{3}[\/latex].<\/p>\r\n<p id=\"fs-id1170572472189\">We now consider a composition of a trigonometric function and its inverse. For example, consider the two expressions [latex] \\sin (\\sin^{-1}(\\frac{\\sqrt{2}}{2}))[\/latex] and [latex]\\sin^{-1}(\\sin(\\pi))[\/latex]. For the first one, we simplify as follows:<\/p>\r\n\r\n<div id=\"fs-id1170572472267\" class=\"equation unnumbered\">[latex] \\sin (\\sin^{-1}(\\frac{\\sqrt{2}}{2}))= \\sin (\\frac{\\pi}{4})=\\frac{\\sqrt{2}}{2}[\/latex].<\/div>\r\nFor the second one, we have\r\n<div id=\"fs-id1170572548623\" class=\"equation unnumbered\">[latex]\\sin^{-1}( \\sin (\\pi))=\\sin^{-1}(0)=0[\/latex].<\/div>\r\n<p id=\"fs-id1170572548676\">The inverse function is supposed to \u201cundo\u201d the original function, so why isn\u2019t [latex]\\sin^{-1}(\\sin (\\pi))=\\pi [\/latex]? Recalling our definition of inverse functions, a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] satisfy the conditions [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in the domain of [latex]f^{-1}[\/latex] and [latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], so what happened here? The issue is that the inverse sine function, [latex]\\sin^{-1}[\/latex], is the inverse of the <em>restricted<\/em> sine function defined on the domain [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex]. Therefore, for [latex]x[\/latex] in the interval [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex], it is true that [latex]\\sin^{-1}(\\sin x)=x[\/latex]. However, for values of [latex]x[\/latex] outside this interval, the equation does not hold, even though [latex]\\sin^{-1}(\\sin x)[\/latex] is defined for all real numbers [latex]x[\/latex].<\/p>\r\n<p id=\"fs-id1170572089863\">What about [latex] \\sin (\\sin^{-1}y)[\/latex]? Does that have a similar issue? The answer is <em>no<\/em>. Since the domain of [latex]\\sin^{-1}[\/latex] is the interval [latex][-1,1][\/latex], we conclude that [latex] \\sin (\\sin^{-1}y)=y[\/latex] if [latex]-1 \\le y \\le 1[\/latex] and the expression is not defined for other values of [latex]y[\/latex]. To summarize,<\/p>\r\n\r\n<div id=\"fs-id1170572550786\" class=\"equation unnumbered\">[latex] \\sin (\\sin^{-1}y)=y \\, \\text{if} \\, -1 \\le y \\le 1[\/latex]<\/div>\r\n<p id=\"fs-id1170572550837\">and<\/p>\r\n\r\n<div id=\"fs-id1170572550840\" class=\"equation unnumbered\">[latex]\\sin^{-1}( \\sin x)=x \\, \\text{if} \\, -\\frac{\\pi}{2} \\le x \\le \\frac{\\pi}{2}[\/latex].<\/div>\r\n<p id=\"fs-id1170572550900\">Similarly, for the cosine function,<\/p>\r\n\r\n<div id=\"fs-id1170572550903\" class=\"equation unnumbered\">[latex] \\cos (\\cos^{-1}y)=y \\, \\text{if} \\, -1 \\le y \\le 1[\/latex]<\/div>\r\n<p id=\"fs-id1170572549577\">and<\/p>\r\n\r\n<div id=\"fs-id1170572549580\" class=\"equation unnumbered\">[latex]\\cos^{-1}( \\cos x)=x \\, \\text{if} 0 \\le x \\le \\pi[\/latex].<\/div>\r\n<p id=\"fs-id1170572549631\">Similar properties hold for the other trigonometric functions and their inverses.<\/p>\r\n\r\n<div id=\"fs-id1170572549634\" class=\"textbox examples\">\r\n<h3>Evaluating Expressions Involving Inverse Trigonometric Functions<\/h3>\r\n<div id=\"fs-id1170572549640\" class=\"exercise\">\r\n<div id=\"fs-id1170572549642\" class=\"textbox\">\r\n<p id=\"fs-id1170572549644\">Evaluate each of the following expressions.<\/p>\r\n\r\n<ol id=\"fs-id1170572549647\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\sin^{-1}(-\\frac{\\sqrt{3}}{2})[\/latex]<\/li>\r\n \t<li>[latex] \\tan (\\tan^{-1}(-\\frac{1}{\\sqrt{3}}))[\/latex]<\/li>\r\n \t<li>[latex]\\cos^{-1}( \\cos (\\frac{5\\pi}{4}))[\/latex]<\/li>\r\n \t<li>[latex]\\sin^{-1}( \\cos (\\frac{2\\pi}{3}))[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572549361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572549361\"]\r\n<ol id=\"fs-id1170572549361\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Evaluating [latex]\\sin^{-1}(\u2212\\sqrt{3}\/2)[\/latex] is equivalent to finding the angle [latex]\\theta [\/latex] such that [latex] \\sin \\theta =\u2212\\sqrt{3}\/2[\/latex] and [latex]\u2212\\pi\/2 \\le \\theta \\le \\pi\/2[\/latex]. The angle [latex]\\theta =\u2212\\pi\/3[\/latex] satisfies these two conditions. Therefore, [latex]\\sin^{-1}(\u2212\\sqrt{3}\/2)=\u2212\\pi\/3[\/latex].<\/li>\r\n \t<li>First we use the fact that [latex]\\tan^{-1}(-1\/\\sqrt{3})=\u2212\\pi\/6[\/latex]. Then [latex] \\tan (\\pi\/6)=-1\/\\sqrt{3}[\/latex]. Therefore, [latex] \\tan (\\tan^{-1}(-1\/\\sqrt{3}))=-1\/\\sqrt{3}[\/latex].<\/li>\r\n \t<li>To evaluate [latex]\\cos^{-1}( \\cos (5\\pi\/4))[\/latex], first use the fact that [latex] \\cos (5\\pi\/4)=\u2212\\sqrt{2}\/2[\/latex]. Then we need to find the angle [latex]\\theta [\/latex] such that [latex] \\cos (\\theta )=\u2212\\sqrt{2}\/2[\/latex] and [latex]0 \\le \\theta \\le \\pi[\/latex]. Since [latex]3\\pi\/4[\/latex] satisfies both these conditions, we have [latex] \\cos (\\cos^{-1}(5\\pi\/4))= \\cos (\\cos^{-1}(\u2212\\sqrt{2}\/2))=3\\pi\/4[\/latex].<\/li>\r\n \t<li>Since [latex] \\cos (2\\pi\/3)=-1\/2[\/latex], we need to evaluate [latex]\\sin^{-1}(-1\/2)[\/latex]. That is, we need to find the angle [latex]\\theta [\/latex] such that [latex] \\sin (\\theta )=-1\/2[\/latex] and [latex]\u2212\\pi\/2 \\le \\theta \\le \\pi\/2[\/latex]. Since [latex]\u2212\\pi\/6[\/latex] satisfies both these conditions, we can conclude that [latex]\\sin^{-1}( \\cos (2\\pi\/3))=\\sin^{-1}(-1\/2)=\u2212\\pi\/6[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572169395\" class=\"textbox key-takeaways project\">\r\n<h3>The Maximum Value of a Function<\/h3>\r\n<p id=\"fs-id1170572169402\">In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don\u2019t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.<\/p>\r\n<p id=\"fs-id1170572169416\">This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable [latex]x[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572169426\">\r\n \t<li>Consider the graph in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_016\">(Figure)<\/a> of the function [latex]y= \\sin x + \\cos x[\/latex]. Describe its overall shape. Is it periodic? How do you know?\r\n<div id=\"CNX_Calc_Figure_01_04_016\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202607\/CNX_Calc_Figure_01_04_016.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of the function \u201cy = sin(x) + cos(x)\u201d, a curved wave function. The graph of the function decreases until it hits the approximate point (-(3pi\/4), -1.4), where it increases until the approximate point ((pi\/4), 1.4), where it begins to decrease again. The x intercepts shown on this graph of the function are at (-(5pi\/4), 0), (-(pi\/4), 0), and ((3pi\/4), 0). The y intercept is at (0, 1).\" width=\"325\" height=\"308\" \/> <strong>Figure 6.<\/strong> The graph of [latex]y= \\sin x + \\cos x[\/latex].[\/caption]<\/div>\r\nUsing a graphing calculator or other graphing device, estimate the [latex]x[\/latex]- and [latex]y[\/latex]-values of the maximum point for the graph (the first such point where [latex]x&gt;0[\/latex]). It may be helpful to express the [latex]x[\/latex]-value as a multiple of [latex]\\pi[\/latex].<\/li>\r\n \t<li>Now consider other graphs of the form [latex]y=A \\sin x + B \\cos x[\/latex] for various values of [latex]A[\/latex] and [latex]B[\/latex]. Sketch the graph when [latex]A = 2[\/latex] and [latex]B = 1[\/latex], and find the [latex]x[\/latex]- and [latex]y[\/latex]-values for the maximum point. (Remember to express the [latex]x[\/latex]-value as a multiple of [latex]\\pi[\/latex], if possible.) Has it moved?<\/li>\r\n \t<li>Repeat for [latex]A = 1, \\, B = 2[\/latex]. Is there any relationship to what you found in part (2)?<\/li>\r\n \t<li>Complete the following table, adding a few choices of your own for [latex]A[\/latex] and [latex]B[\/latex]:\r\n<table id=\"fs-id1170572554057\" class=\"unnumbered\" summary=\"A table containing 8 columns and 9 rows is shown. The first column is labeled \u201cA\u201d and contains the values \u201c0,1,1,1,2,2,3, and 4.\u201d The second column is labeled \u201cB\u201d and contains the values \u201c1,0,1,2,1,2,4, and 3.\u201d The third column is labeled \u201cx\u201d and has no values for any of the rows. The fourth column is labeled \u201cy\u201d and contains no values for any of the rows. The fifth column is separated from the fourth column by a gutter, is labeled \u201cA\u201d and contains the values \u201cthe square root of 3,1,12, and 5.\u201d The sixth column is labeled \u201cB\u201d and contains the values 1, the square root of 3,5, and 12.\u201d The seventh column is labeled \u201cx\u201d and contains no values for any of the rows. The eighth column is labeled \u201cy\u201d and contains no values for any of the rows.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]A[\/latex]<\/th>\r\n<th>[latex]B[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]A[\/latex]<\/th>\r\n<th>[latex]B[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td rowspan=\"8\"><\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>12<\/td>\r\n<td>5<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>5<\/td>\r\n<td>12<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>2<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>4<\/td>\r\n<td>3<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Try to figure out the formula for the [latex]y[\/latex]-values.<\/li>\r\n \t<li>The formula for the [latex]x[\/latex]-values is a little harder. The most helpful points from the table are [latex](1,1), \\, (1,\\sqrt{3}), \\, (\\sqrt{3},1)[\/latex]. (<em>Hint<\/em>: <em>Consider inverse trigonometric functions.)<\/em><\/li>\r\n \t<li>If you found formulas for parts (5) and (6), show that they work together. That is, substitute the [latex]x[\/latex]-value formula you found into [latex]y=A \\sin x + B \\cos x[\/latex] and simplify it to arrive at the [latex]y[\/latex]-value formula you found.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572470426\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1170572470433\">\r\n \t<li>For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.<\/li>\r\n \t<li>If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.<\/li>\r\n \t<li>For a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f^{-1}[\/latex] and [latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\r\n \t<li>Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.<\/li>\r\n \t<li>The graph of a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] are symmetric about the line [latex]y=x[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572453118\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<ul id=\"fs-id1170572453126\">\r\n \t<li><strong>Inverse functions<\/strong>\r\n[latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in [latex]D[\/latex], and [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in [latex]R[\/latex].[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572547960\" class=\"textbox exercises\">\r\n<p id=\"fs-id1170572547964\">For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.<\/p>\r\n\r\n<div id=\"fs-id1170572547969\" class=\"exercise\">\r\n<div id=\"fs-id1170572547971\" class=\"textbox\"><span id=\"fs-id1170572547973\"><strong>1.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202609\/CNX_Calc_Figure_01_04_201.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that decreases in a straight in until the origin, where it begins to increase in a straight line. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572547990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572547990\"]\r\n<p id=\"fs-id1170572547990\">Not one-to-one<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572547995\" class=\"exercise\">\r\n<div id=\"fs-id1170572547997\" class=\"textbox\"><span id=\"fs-id1170572547999\"><strong>2.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202612\/CNX_Calc_Figure_01_04_202.jpg\" alt=\"An image of a graph. The x axis runs from 0 to 7 and the y axis runs from -4 to 4. The graph is of a function that is always increasing. There is an approximate x intercept at the point (1, 0) and no y intercept shown.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548022\" class=\"exercise\">\r\n<div id=\"fs-id1170572548024\" class=\"textbox\"><span id=\"fs-id1170572548026\"><strong>3.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202614\/CNX_Calc_Figure_01_04_203.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that resembles a semi-circle, the top half of a circle. The function starts at the point (-3, 0) and increases until the point (0, 3), where it begins decreasing until it ends at the point (3, 0). The x intercepts are at (-3, 0) and (3, 0). The y intercept is at (0, 3).\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572548044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572548044\"]\r\n<p id=\"fs-id1170572548044\">Not one-to-one<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548050\" class=\"exercise\">\r\n<div id=\"fs-id1170572548052\" class=\"textbox\"><span id=\"fs-id1170572548054\"><strong>4.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202617\/CNX_Calc_Figure_01_04_204.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function. The function increases until it hits the origin, then decreases until it hits the point (2, -4), where it begins to increase again. There are x intercepts at the origin and the point (3, 0). The y intercept is at the origin.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548077\" class=\"exercise\">\r\n<div id=\"fs-id1170572548079\" class=\"textbox\"><span id=\"fs-id1170572548081\"><strong>5.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202620\/CNX_Calc_Figure_01_04_205.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function that is always increasing. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572548098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572548098\"]\r\n<p id=\"fs-id1170572548098\">One-to-one<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452097\" class=\"exercise\">\r\n<div id=\"fs-id1170572452099\" class=\"textbox\"><span id=\"fs-id1170572452101\"><strong>6.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202622\/CNX_Calc_Figure_01_04_206.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 7 and the y axis runs from -4 to 4. The graph is of a function that increases in a straight line until the approximate point (, 3). After this point, the function becomes a horizontal straight line. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572452124\">For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.<\/p>\r\n\r\n<div id=\"fs-id1170572452128\" class=\"exercise\">\r\n<div id=\"fs-id1170572452130\" class=\"textbox\">\r\n<p id=\"fs-id1170572452132\"><strong>7.\u00a0<\/strong>[latex]f(x)=x^2-4, \\, x \\ge 0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572452170\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452170\"]\r\n<p id=\"fs-id1170572452170\">a. [latex]f^{-1}(x)=\\sqrt{x+4}[\/latex] b. Domain: [latex]x \\ge -4[\/latex], Range: [latex]y \\ge 0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452229\" class=\"exercise\">\r\n<div id=\"fs-id1170572452231\" class=\"textbox\">\r\n<p id=\"fs-id1170572452233\"><strong>8.\u00a0<\/strong>[latex]f(x)=\\sqrt[3]{x-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572176863\" class=\"exercise\">\r\n<div id=\"fs-id1170572176865\" class=\"textbox\">\r\n<p id=\"fs-id1170572176867\"><strong>9.\u00a0<\/strong>[latex]f(x)=x^3+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572176896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572176896\"]\r\n<p id=\"fs-id1170572176896\">a. [latex]f^{-1}(x)=\\sqrt[3]{x-1}[\/latex] b. Domain: all real numbers, Range: all real numbers<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572176931\" class=\"exercise\">\r\n<div id=\"fs-id1170572176934\" class=\"textbox\">\r\n<p id=\"fs-id1170572176936\"><strong>10.\u00a0<\/strong>[latex]f(x)=(x-1)^2, \\, x \\le 1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572549738\" class=\"exercise\">\r\n<div id=\"fs-id1170572549740\" class=\"textbox\">\r\n<p id=\"fs-id1170572549742\"><strong>11.\u00a0<\/strong>[latex]f(x)=\\sqrt{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572549770\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572549770\"]\r\n<p id=\"fs-id1170572549770\">a. [latex]f^{-1}(x)=x^2+1[\/latex], b. Domain: [latex]x \\ge 0[\/latex], Range: [latex]y \\ge 1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572549826\" class=\"exercise\">\r\n<div id=\"fs-id1170572549828\" class=\"textbox\">\r\n<p id=\"fs-id1170572549831\"><strong>12.\u00a0<\/strong>[latex]f(x)=\\frac{1}{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572451519\">For the following exercises, use the graph of [latex]f[\/latex] to sketch the graph of its inverse function.<\/p>\r\n\r\n<div id=\"fs-id1170572451526\" class=\"exercise\">\r\n<div id=\"fs-id1170572451528\" class=\"textbox\"><span id=\"fs-id1170572451534\"><strong>13.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202625\/CNX_Calc_Figure_01_04_207.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of an increasing straight line function labeled \u201cf\u201d that is always increasing. The x intercept is at (-2, 0) and y intercept are both at (0, 1).\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572451548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572451548\"]<span id=\"fs-id1170572451558\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202627\/CNX_Calc_Figure_01_04_208.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of two functions. The first function is an increasing straight line function labeled \u201cf\u201d. The x intercept is at (-2, 0) and y intercept are both at (0, 1). The second function is of an increasing straight line function labeled \u201cf inverse\u201d. The x intercept is at the point (1, 0) and the y intercept is at the point (0, -2).\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572451569\" class=\"exercise\">\r\n<div id=\"fs-id1170572451571\" class=\"textbox\"><span id=\"fs-id1170572451577\"><strong>14.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202630\/CNX_Calc_Figure_01_04_209.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved decreasing function labeled \u201cf\u201d. As the function decreases, it gets approaches the x axis but never touches it. The function does not have an x intercept and the y intercept is (0, 1).\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452480\" class=\"exercise\">\r\n<div id=\"fs-id1170572452482\" class=\"textbox\"><span id=\"fs-id1170572452491\"><strong>15.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202633\/CNX_Calc_Figure_01_04_211.jpg\" alt=\"An image of a graph. The x axis runs from -8 to 8 and the y axis runs from -8 to 8. The graph is of an increasing straight line function labeled \u201cf\u201d. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1).\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572452503\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452503\"]<span id=\"fs-id1170572452513\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202636\/CNX_Calc_Figure_01_04_212.jpg\" alt=\"An image of a graph. The x axis runs from 0 to 8 and the y axis runs from 0 to 8. The graph is of two function. The first function is an increasing straight line function labeled \u201cf\u201d. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1). The second function is an increasing straight line function labeled \u201cf inverse\u201d. The function starts at the point (1, 0) and increases in straight line until the point (6, 4). After this point, the function continues to increase, but at a faster rate than before, as it approaches the point (8, 8). The function does not have an y intercept and the x intercept is (1, 0).\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452528\" class=\"exercise\">\r\n<div id=\"fs-id1170572452530\" class=\"textbox\"><span id=\"fs-id1170572452536\"><strong>16.\r\n<\/strong><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202639\/CNX_Calc_Figure_01_04_213.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a decreasing curved function labeled \u201cf\u201d, which ends at the origin, which is both the x intercept and y intercept. Another point on the function is (-4, 2).\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572452571\">For the following exercises, use composition to determine which pairs of functions are inverses.<\/p>\r\n\r\n<div id=\"fs-id1170572452575\" class=\"exercise\">\r\n<div id=\"fs-id1170572452577\" class=\"textbox\">\r\n<p id=\"fs-id1170572452579\"><strong>17.\u00a0<\/strong>[latex]f(x)=8x, \\, g(x)=\\frac{x}{8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572452622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452622\"]\r\n<p id=\"fs-id1170572452622\">These are inverses.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572452632\"><strong>18.\u00a0<\/strong>[latex]f(x)=8x+3, \\, g(x)=\\frac{x-3}{8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572449207\" class=\"exercise\">\r\n<div id=\"fs-id1170572449209\" class=\"textbox\">\r\n<p id=\"fs-id1170572449211\"><strong>19.\u00a0<\/strong>[latex]f(x)=5x-7, \\, g(x)=\\frac{x+5}{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572449263\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572449263\"]\r\n<p id=\"fs-id1170572449263\">These are not inverses.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572449269\" class=\"exercise\">\r\n<div id=\"fs-id1170572449271\" class=\"textbox\">\r\n<p id=\"fs-id1170572449273\"><strong>20.\u00a0<\/strong>[latex]f(x)=\\frac{2}{3}x+2, \\, g(x)=\\frac{3}{2}x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548356\" class=\"exercise\">\r\n<div id=\"fs-id1170572548358\" class=\"textbox\">\r\n<p id=\"fs-id1170572548360\"><strong>21.\u00a0<\/strong>[latex]f(x)=\\frac{1}{x-1}, \\, x \\ne 1, \\, g(x)=\\frac{1}{x}+1, \\, x \\ne 0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572548430\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572548430\"]\r\n<p id=\"fs-id1170572548430\">These are inverses.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548436\" class=\"exercise\">\r\n<div id=\"fs-id1170572548438\" class=\"textbox\">\r\n<p id=\"fs-id1170572548440\"><strong>22. <\/strong>[latex]f(x)=x^3+1, \\, g(x)=(x-1)^{1\/3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548511\" class=\"exercise\">\r\n<div id=\"fs-id1170572229236\" class=\"textbox\">\r\n<p id=\"fs-id1170572229238\"><strong>23.\u00a0<\/strong>[latex]f(x)=x^2+2x+1, \\, x \\ge -1,\\, g(x)=-1+\\sqrt{x}, \\, x \\ge 0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572229326\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572229326\"]\r\n<p id=\"fs-id1170572229326\">These are inverses.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572229332\" class=\"exercise\">\r\n<div id=\"fs-id1170572229334\" class=\"textbox\">\r\n<p id=\"fs-id1170572229336\"><strong>24.\u00a0<\/strong>[latex]f(x)=\\sqrt{4-x^2}, \\, 0 \\le x \\le 2, \\, g(x)=\\sqrt{4-x^2}, \\, 0 \\le x \\le 2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572451285\">For the following exercises, evaluate the functions. Give the exact value.<\/p>\r\n\r\n<div id=\"fs-id1170572451288\" class=\"exercise\">\r\n<div id=\"fs-id1170572451291\" class=\"textbox\">\r\n<p id=\"fs-id1170572451293\"><strong>25.\u00a0<\/strong>[latex]\\tan^{-1}(\\frac{\\sqrt{3}}{3})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572451322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572451322\"]\r\n<p id=\"fs-id1170572451322\">[latex]\\frac{\\pi}{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572451334\" class=\"exercise\">\r\n<div id=\"fs-id1170572451336\" class=\"textbox\">\r\n\r\n<strong>26.\u00a0<\/strong>[latex]\\cos^{-1}(-\\frac{\\sqrt{2}}{2})[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572451384\" class=\"exercise\">\r\n<div id=\"fs-id1170572451386\" class=\"textbox\">\r\n<p id=\"fs-id1170572451388\"><strong>27.\u00a0<\/strong>[latex]\\cot^{-1}(1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572451411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572451411\"]\r\n<p id=\"fs-id1170572451411\">[latex]\\frac{\\pi}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572451423\" class=\"exercise\">\r\n<div id=\"fs-id1170572451425\" class=\"textbox\">\r\n<p id=\"fs-id1170572451427\"><strong>28.\u00a0<\/strong>[latex]\\sin^{-1}(-1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572142146\" class=\"exercise\">\r\n<div id=\"fs-id1170572142148\" class=\"textbox\">\r\n<p id=\"fs-id1170572142150\"><strong>29.\u00a0<\/strong>[latex]\\cos^{-1}(\\frac{\\sqrt{3}}{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572142179\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572142179\"]\r\n<p id=\"fs-id1170572142179\">[latex]\\frac{\\pi}{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572142191\" class=\"exercise\">\r\n<div id=\"fs-id1170572142193\" class=\"textbox\">\r\n<p id=\"fs-id1170572142195\"><strong>30.\u00a0<\/strong>[latex] \\cos (\\tan^{-1}(\\sqrt{3}))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572142240\" class=\"exercise\">\r\n<div id=\"fs-id1170572142242\" class=\"textbox\">\r\n<p id=\"fs-id1170572142245\"><strong>31.\u00a0<\/strong>[latex] \\sin (\\cos^{-1}(\\frac{\\sqrt{2}}{2}))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">[reveal-answer q=\"461959\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"461959\"][latex]\\frac{\\sqrt{2}}{2}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572142296\" class=\"exercise\">\r\n<div id=\"fs-id1170572548960\" class=\"textbox\">\r\n<p id=\"fs-id1170572548962\"><strong>32.\u00a0<\/strong>[latex]\\sin^{-1}( \\sin (\\frac{\\pi}{3}))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572549009\" class=\"exercise\">\r\n<div id=\"fs-id1170572549011\" class=\"textbox\">\r\n<p id=\"fs-id1170572549013\"><strong>33.\u00a0<\/strong>[latex]\\tan^{-1}( \\tan (-\\frac{\\pi}{6}))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572549051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572549051\"]\r\n<p id=\"fs-id1170572549051\">[latex]-\\frac{\\pi}{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572549065\" class=\"exercise\">\r\n<div id=\"fs-id1170572549067\" class=\"textbox\">\r\n<p id=\"fs-id1170572549069\"><strong>34.\u00a0<\/strong>The function [latex]C=T(F)=(5\/9)(F-32)[\/latex] converts degrees Fahrenheit to degrees Celsius.<\/p>\r\n\r\n<ol id=\"fs-id1170572549118\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the inverse function [latex]F=T^{-1}(C)[\/latex]<\/li>\r\n \t<li>What is the inverse function used for?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572451744\" class=\"exercise\">\r\n<div id=\"fs-id1170572451746\" class=\"textbox\">\r\n<p id=\"fs-id1170572451748\"><strong>35. [T]<\/strong> The velocity [latex]V[\/latex] (in centimeters per second) of blood in an artery at a distance [latex]x[\/latex] cm from the center of the artery can be modeled by the function [latex]V=f(x)=500(0.04-x^2)[\/latex] for [latex]0 \\le x \\le 0.2[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572451823\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find [latex]x=f^{-1}(V)[\/latex].<\/li>\r\n \t<li>Interpret what the inverse function is used for.<\/li>\r\n \t<li>Find the distance from the center of an artery with a velocity of 15 cm\/sec, 10 cm\/sec, and 5 cm\/sec.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572547753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572547753\"]\r\n<p id=\"fs-id1170572547753\">a. [latex]x=f^{-1}(V)=\\sqrt{0.04-\\frac{V}{500}}[\/latex] b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity [latex]V[\/latex]. c. 0.1 cm; 0.14 cm; 0.17 cm<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572547801\" class=\"exercise\">\r\n<div id=\"fs-id1170572547803\" class=\"textbox\">\r\n<p id=\"fs-id1170572547805\"><strong>36.\u00a0<\/strong>A function that converts dress sizes in the United States to those in Europe is given by [latex]D(x)=2x+24[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572547832\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States.<\/li>\r\n \t<li>Find the function that converts European dress sizes to U.S. dress sizes.<\/li>\r\n \t<li>Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572547878\" class=\"exercise\">\r\n<div id=\"fs-id1170572547880\" class=\"textbox\">\r\n<p id=\"fs-id1170572547882\"><strong>37. [T]<\/strong> The cost to remove a toxin from a lake is modeled by the function<\/p>\r\n<p id=\"fs-id1170572547889\">[latex]C(p)=75p\/(85-p)[\/latex], where [latex]C[\/latex] is the cost (in thousands of dollars) and [latex]p[\/latex] is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.<\/p>\r\n\r\n<ol id=\"fs-id1170572542869\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.<\/li>\r\n \t<li>Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572542887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572542887\"]\r\n<p id=\"fs-id1170572542887\">a. $31,250, $66,667, $107,143 b. [latex](p=\\frac{85C}{C+75})[\/latex] c. 34 ppb<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572542921\" class=\"exercise\">\r\n<div id=\"fs-id1170572542923\" class=\"textbox\">\r\n<p id=\"fs-id1170572542925\"><strong>38. [T]<\/strong> A race car is accelerating at a velocity given by<\/p>\r\n<p id=\"fs-id1170572542932\">[latex]v(t)=\\frac{25}{4}t+54[\/latex],<\/p>\r\n<p id=\"fs-id1170572542966\">where [latex]v[\/latex] is the velocity (in feet per second) at time [latex]t[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572542980\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the velocity of the car at 10 sec.<\/li>\r\n \t<li>Find the inverse function.<\/li>\r\n \t<li>Use part b. to determine how long it takes for the car to reach a speed of 150 ft\/sec.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572543027\" class=\"exercise\">\r\n<div id=\"fs-id1170572455080\" class=\"textbox\">\r\n<p id=\"fs-id1170572455082\"><strong>39. [T]<\/strong> An airplane\u2019s Mach number [latex]M[\/latex] is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by [latex]\\mu =2\\sin^{-1}(\\frac{1}{M})[\/latex].<\/p>\r\n<p id=\"fs-id1170572455127\">Find the Mach angle (to the nearest degree) for the following Mach numbers.<\/p>\r\n<span id=\"fs-id1170572455137\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202641\/CNX_Calc_Figure_01_04_215.jpg\" alt=\"An image of a birds eye view of an airplane. Directly in front of the airplane is a sideways \u201cV\u201d shape, with the airplane flying directly into the opening of the \u201cV\u201d shape. The \u201cV\u201d shape is labeled \u201cmach wave\u201d. There are two arrows with labels. The first arrow points from the nose of the airplane to the corner of the \u201cV\u201d shape. This arrow has the label \u201cvelocity = v\u201d. The second arrow points diagonally from the nose of the airplane to the edge of the upper portion of the \u201cV\u201d shape. This arrow has the label \u201cspeed of sound = a\u201d. Between these two arrows is an angle labeled \u201cMach angle\u201d. There is also text in the image that reads \u201cmach = M &gt; 1.0\u201d.\" \/><\/span>\r\n<ol id=\"fs-id1170572455148\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\mu =1.4[\/latex]<\/li>\r\n \t<li>[latex]\\mu =2.8[\/latex]<\/li>\r\n \t<li>[latex]\\mu =4.3[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572455191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572455191\"]\r\n<p id=\"fs-id1170572455191\">a. [latex]~92^{\\circ}[\/latex] b. [latex]~42^{\\circ}[\/latex] c. [latex]~27^{\\circ}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572455224\" class=\"exercise\">\r\n<div id=\"fs-id1170572455226\" class=\"textbox\">\r\n<p id=\"fs-id1170572455228\"><strong>40. [T]<\/strong> Using [latex]\\mu =2\\sin^{-1}(\\frac{1}{M})[\/latex], find the Mach number [latex]M[\/latex] for the following angles.<\/p>\r\n\r\n<ol id=\"fs-id1170572551412\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\mu =\\frac{\\pi}{6}[\/latex]<\/li>\r\n \t<li>[latex]\\mu =\\frac{2\\pi}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\mu =\\frac{3\\pi}{8}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572551494\" class=\"exercise\">\r\n<div id=\"fs-id1170572551497\" class=\"textbox\">\r\n<p id=\"fs-id1170572551499\"><strong>41. [T]<\/strong> The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function<\/p>\r\n<p id=\"fs-id1170572551507\">[latex]T(x)=5+18 \\sin[\\frac{\\pi}{6}(x-4.6)][\/latex],<\/p>\r\n<p id=\"fs-id1170572551558\">where [latex]x[\/latex] is time in months and [latex]x=1.00[\/latex] corresponds to January 1. Determine the month and day when the temperature is [latex]21^{\\circ}[\/latex] C.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572545094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572545094\"]\r\n<p id=\"fs-id1170572545094\">[latex]x \\approx 6.69,8.51[\/latex]; so, the temperature occurs on June 21 and August 15<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572545115\" class=\"exercise\">\r\n<div id=\"fs-id1170572545117\" class=\"textbox\">\r\n<p id=\"fs-id1170572545119\"><strong>42. [T]<\/strong> The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function<\/p>\r\n<p id=\"fs-id1170572545127\">[latex]D(t)=5 \\sin (\\frac{\\pi}{6}t-\\frac{7\\pi}{6})+8[\/latex],<\/p>\r\n<p id=\"fs-id1170572545179\">where [latex]t[\/latex] is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572545202\" class=\"exercise\">\r\n<div id=\"fs-id1170572545204\" class=\"textbox\">\r\n<p id=\"fs-id1170572545207\"><strong>43. [T]<\/strong> An object moving in simple harmonic motion is modeled by the function<\/p>\r\n<p id=\"fs-id1170572545214\">[latex]s(t)=-6 \\cos (\\frac{\\pi t}{2})[\/latex],<\/p>\r\n<p id=\"fs-id1170572545252\">where [latex]s[\/latex] is measured in inches and [latex]t[\/latex] is measured in seconds. Determine the first time when the distance moved is 4.5 ft.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572545265\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572545265\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572545265\"][latex]~1.5 \\sec [\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572169073\" class=\"exercise\">\r\n<div id=\"fs-id1170572169076\" class=\"textbox\">\r\n<p id=\"fs-id1170572169078\"><strong>44. [T]<\/strong> A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle [latex]\\theta [\/latex] can be modeled by the function<\/p>\r\n<p id=\"fs-id1170572169091\">[latex]\\theta =\\tan^{-1}\\frac{5.5}{x}-\\tan^{-1}\\frac{2.5}{x}[\/latex],<\/p>\r\n<p id=\"fs-id1170572169132\">where [latex]x[\/latex] is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572169169\" class=\"exercise\">\r\n<div id=\"fs-id1170572169171\" class=\"textbox\">\r\n<p id=\"fs-id1170572169173\"><strong>45. [T]<\/strong> Use a calculator to evaluate [latex]\\tan^{-1}( \\tan (2.1))[\/latex] and [latex]\\cos^{-1}( \\cos (2.1))[\/latex]. Explain the results of each.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572243719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572243719\"]\r\n<p id=\"fs-id1170572243719\">[latex]\\tan^{-1}( \\tan (2.1))\\approx -1.0416[\/latex]; the expression does not equal 2.1 since [latex]2.1&gt;1.57=\\frac{\\pi}{2}[\/latex]\u2014in other words, it is not in the restricted domain of [latex] \\tan x[\/latex].\u00a0 [latex]\\cos^{-1}( \\cos (2.1))=2.1[\/latex], since 2.1 is in the restricted domain of [latex] \\cos x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572243833\" class=\"exercise\">\r\n<div id=\"fs-id1170572243835\" class=\"textbox\">\r\n\r\n<strong>46. [T]<\/strong> Use a calculator to evaluate [latex] \\sin (\\sin^{-1}(-2))[\/latex] and [latex] \\tan (\\tan^{-1}(-2))[\/latex]. Explain the results of each.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572229168\" class=\"definition\">\r\n \t<dt>horizontal line test<\/dt>\r\n \t<dd id=\"fs-id1170572229174\">a function [latex]f[\/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[\/latex], at most, once<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572229190\" class=\"definition\">\r\n \t<dt>inverse function<\/dt>\r\n \t<dd id=\"fs-id1170572229195\">for a function [latex]f[\/latex], the inverse function [latex]f^{-1}[\/latex] satisfies [latex]f^{-1}(y)=x[\/latex] if [latex]f(x)=y[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572482608\" class=\"definition\">\r\n \t<dt>inverse trigonometric functions<\/dt>\r\n \t<dd id=\"fs-id1170572482614\">the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572482619\" class=\"definition\">\r\n \t<dt>one-to-one function<\/dt>\r\n \t<dd id=\"fs-id1170572482624\">a function [latex]f[\/latex] is one-to-one if [latex]f(x_1) \\ne f(x_2)[\/latex] if [latex]x_1 \\ne x_2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572482683\" class=\"definition\">\r\n \t<dt>restricted domain<\/dt>\r\n \t<dd id=\"fs-id1170572482689\">a subset of the domain of a function [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Determine the conditions for when a function has an inverse.<\/li>\n<li>Use the horizontal line test to recognize when a function is one-to-one.<\/li>\n<li>Find the inverse of a given function.<\/li>\n<li>Draw the graph of an inverse function.<\/li>\n<li>Evaluate inverse trigonometric functions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170572215818\">An <strong>inverse function<\/strong> reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. In this section, we define an inverse function formally and state the necessary conditions for an inverse function to exist. We examine how to find an inverse function and study the relationship between the graph of a function and the graph of its inverse. Then we apply these ideas to define and discuss properties of the inverse trigonometric functions.<\/p>\n<div id=\"fs-id1170572245927\" class=\"bc-section section\">\n<h1>Existence of an Inverse Function<\/h1>\n<p>We begin with an example. Given a function [latex]f[\/latex] and an output [latex]y=f(x)[\/latex], we are often interested in finding what value or values [latex]x[\/latex] were mapped to [latex]y[\/latex] by [latex]f[\/latex]. For example, consider the function [latex]f(x)=x^3+4[\/latex]. Since any output [latex]y=x^3+4[\/latex], we can solve this equation for [latex]x[\/latex] to find that the input is [latex]x=\\sqrt[3]{y-4}[\/latex]. This equation defines [latex]x[\/latex] as a function of [latex]y[\/latex]. Denoting this function as [latex]f^{-1}[\/latex], and writing [latex]x=f^{-1}(y)=\\sqrt[3]{y-4}[\/latex], we see that for any [latex]x[\/latex] in the domain of [latex]f, \\, f^{-1}(f(x))=f^{-1}(x^3+4)=x[\/latex]. Thus, this new function, [latex]f^{-1}[\/latex], \u201cundid\u201d what the original function [latex]f[\/latex] did. A function with this property is called the inverse function of the original function.<\/p>\n<div id=\"fs-id1170572137845\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1170572110489\">Given a function [latex]f[\/latex] with domain [latex]D[\/latex] and range [latex]R[\/latex], its inverse function (if it exists) is the function [latex]f^{-1}[\/latex] with domain [latex]R[\/latex] and range [latex]D[\/latex] such that [latex]f^{-1}(y)=x[\/latex] if [latex]f(x)=y[\/latex]. In other words, for a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex],<\/p>\n<div id=\"fs-id1170572141883\" class=\"equation\">[latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in [latex]D[\/latex], and [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in [latex]R[\/latex].<\/div>\n<\/div>\n<p id=\"fs-id1170572548323\">Note that [latex]f^{-1}[\/latex] is read as \u201cf inverse.\u201d Here, the -1 is not used as an exponent and [latex]f^{-1}(x) \\ne 1\/f(x)[\/latex].\u00a0<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_001\">(Figure)<\/a> shows the relationship between the domain and range of [latex]f[\/latex] and the domain and range of [latex]f^{-1}[\/latex].<\/p>\n<div id=\"CNX_Calc_Figure_01_04_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 497px\" 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\/11202523\/CNX_Calc_Figure_01_04_001.jpg\" alt=\"An image of two bubbles. The first bubble is orange and has two labels: the top label is \u201cDomain of f\u201d and the bottom label is \u201cRange of f inverse\u201d. Within this bubble is the variable \u201cx\u201d. An orange arrow with the label \u201cf\u201d points from this bubble to the second bubble. The second bubble is blue and has two labels: the top label is \u201crange of f\u201d and the bottom label is \u201cdomain of f inverse\u201d. Within this bubble is the variable \u201cy\u201d. A blue arrow with the label \u201cf inverse\u201d points from this bubble to the first bubble.\" width=\"487\" height=\"160\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Given a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f^{-1}(y)=x[\/latex] if and only if [latex]f(x)=y[\/latex]. The range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex] and the domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572141586\">Recall that a function has exactly one output for each input. Therefore, to define an inverse function, we need to map each input to exactly one output. For example, let\u2019s try to find the inverse function for [latex]f(x)=x^2[\/latex]. Solving the equation [latex]y=x^2[\/latex] for [latex]x[\/latex], we arrive at the equation [latex]x= \\pm \\sqrt{y}[\/latex]. This equation does not describe [latex]x[\/latex] as a function of [latex]y[\/latex] because there are two solutions to this equation for every [latex]y>0[\/latex]. The problem with trying to find an inverse function for [latex]f(x)=x^2[\/latex] is that two inputs are sent to the same output for each output [latex]y>0[\/latex]. The function [latex]f(x)=x^3+4[\/latex] discussed earlier did not have this problem. For that function, each input was sent to a different output. A function that sends each input to a <em>different<\/em> output is called a<strong> one-to-one function.<\/strong><\/p>\n<div id=\"fs-id1170572110310\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1170572229709\">We say a [latex]f[\/latex] is a one-to-one function if [latex]f(x_1) \\ne f(x_2)[\/latex] when [latex]x_1 \\ne x_2[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1170572103424\">One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the [latex]xy[\/latex]-plane, according to the <strong>horizontal line test<\/strong>, it cannot intersect the graph more than once. We note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_002\">(Figure)<\/a>).<\/p>\n<div id=\"fs-id1170572226107\" class=\"textbox key-takeaways\">\n<h3>Rule: Horizontal Line Test<\/h3>\n<p id=\"fs-id1170572137262\">A function [latex]f[\/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[\/latex] no more than once.<\/p>\n<\/div>\n<div id=\"CNX_Calc_Figure_01_04_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 497px\" 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\/11202527\/CNX_Calc_Figure_01_04_002.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -3 to 4. The first graph is of the function \u201cf(x) = x squared\u201d, which is a parabola. The function decreases until it hits the origin, where it begins to increase. The x intercept and y intercept are both at the origin. There are two orange horizontal lines also plotted on the graph, both of which run through the function at two points each. The second graph is of the function \u201cf(x) = x cubed\u201d, which is an increasing curved function. The x intercept and y intercept are both at the origin. There are three orange lines also plotted on the graph, each of which only intersects the function at one point.\" width=\"487\" height=\"313\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong> (a) The function [latex]f(x)=x^2[\/latex] is not one-to-one because it fails the horizontal line test. (b) The function [latex]f(x)=x^3[\/latex] is one-to-one because it passes the horizontal line test.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572228079\" class=\"textbox examples\">\n<h3>Determining Whether a Function Is One-to-One<\/h3>\n<div id=\"fs-id1170572102492\" class=\"exercise\">\n<div id=\"fs-id1170572103254\" class=\"textbox\">\n<p id=\"fs-id1170572150756\">For each of the following functions, use the horizontal line test to determine whether it is one-to-one.<\/p>\n<ol id=\"fs-id1170572174892\" style=\"list-style-type: lower-alpha\">\n<li><span id=\"fs-id1170572108141\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202531\/CNX_Calc_Figure_01_04_003.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9)\" \/><\/span><\/li>\n<li><span id=\"fs-id1170572140980\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202534\/CNX_Calc_Figure_01_04_004.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote.\" \/><\/span><\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572241370\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572241370\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572241370\" style=\"list-style-type: lower-alpha\">\n<li>Since the horizontal line [latex]y=n[\/latex] for any integer [latex]n\\ge 0[\/latex] intersects the graph more than once, this function is not one-to-one.<br \/>\n<span id=\"fs-id1170572107320\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202536\/CNX_Calc_Figure_01_04_005.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9). There are also two horizontal orange lines plotted on the graph, each of which run through an entire step of the function.\" \/><\/span><\/li>\n<li>Since every horizontal line intersects the graph once (at most), this function is one-to-one.<br \/>\n<span id=\"fs-id1170572454307\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202538\/CNX_Calc_Figure_01_04_006.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote. There are also three horizontal orange lines plotted on the graph, each of which only runs through the function at one point.\" \/><\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572151745\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572141532\" class=\"exercise\">\n<div id=\"fs-id1170572479279\" class=\"textbox\">\n<p id=\"fs-id1170572294776\">Is the function [latex]f[\/latex] graphed in the following image one-to-one?<\/p>\n<p><span id=\"fs-id1170572224899\"><img 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.\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572216730\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572216730\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572216730\">No.<\/p>\n<\/div>\n<div id=\"fs-id1165042132352\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042094151\">Use the horizontal line test.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572177789\" class=\"bc-section section\">\n<h1>Finding a Function\u2019s Inverse<\/h1>\n<p id=\"fs-id1170572150503\">We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of [latex]f[\/latex] to elements in the range of [latex]f[\/latex]. The inverse function maps each element from the range of [latex]f[\/latex] back to its corresponding element from the domain of [latex]f[\/latex]. Therefore, to find the inverse function of a one-to-one function [latex]f[\/latex], given any [latex]y[\/latex] in the range of [latex]f[\/latex], we need to determine which [latex]x[\/latex] in the domain of [latex]f[\/latex] satisfies [latex]f(x)=y[\/latex]. Since [latex]f[\/latex] is one-to-one, there is exactly one such value [latex]x[\/latex]. We can find that value [latex]x[\/latex] by solving the equation [latex]f(x)=y[\/latex] for [latex]x[\/latex]. Doing so, we are able to write [latex]x[\/latex] as a function of [latex]y[\/latex] where the domain of this function is the range of [latex]f[\/latex] and the range of this new function is the domain of [latex]f[\/latex]. Consequently, this function is the inverse of [latex]f[\/latex], and we write [latex]x=f^{-1}(y)[\/latex]. Since we typically use the variable [latex]x[\/latex] to denote the independent variable and [latex]y[\/latex] to denote the dependent variable, we often interchange the roles of [latex]x[\/latex] and [latex]y[\/latex], and write [latex]y=f^{-1}(x)[\/latex]. Representing the inverse function in this way is also helpful later when we graph a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] on the same axes.<\/p>\n<div id=\"fs-id1170572552427\" class=\"textbox key-takeaways problem-solving\">\n<h3>Problem-Solving Strategy: Finding an Inverse Function<\/h3>\n<ol id=\"fs-id1170572450945\">\n<li>Solve the equation [latex]y=f(x)[\/latex] for [latex]x[\/latex].<\/li>\n<li>Interchange the variables [latex]x[\/latex] and [latex]y[\/latex] and write [latex]y=f^{-1}(x)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572208522\" class=\"textbox examples\">\n<h3>Finding an Inverse Function<\/h3>\n<div id=\"fs-id1170572240528\" class=\"exercise\">\n<div id=\"fs-id1170572546518\" class=\"textbox\">\n<p id=\"fs-id1170572546520\">Find the inverse for the function [latex]f(x)=3x-4[\/latex]. State the domain and range of the inverse function. Verify that [latex]f^{-1}(f(x))=x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572481500\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572481500\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572481500\">Follow the steps outlined in the strategy.<\/p>\n<p id=\"fs-id1170572548820\">Step 1. If [latex]y=3x-4[\/latex], then [latex]3x=y+4[\/latex] and [latex]x=\\frac{1}{3}y+\\frac{4}{3}[\/latex].<\/p>\n<p id=\"fs-id1170572479776\">Step 2. Rewrite as [latex]y=\\frac{1}{3}x+\\frac{4}{3}[\/latex] and let [latex]y=f^{-1}(x)[\/latex].<\/p>\n<p id=\"fs-id1170572453043\">Therefore, [latex]f^{-1}(x)=\\frac{1}{3}x+\\frac{4}{3}[\/latex].<\/p>\n<p id=\"fs-id1170572240611\">Since the domain of [latex]f[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex], the range of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex]. Since the range of [latex]f[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex], the domain of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex].<\/p>\n<p id=\"fs-id1170572222969\">You can verify that [latex]f^{-1}(f(x))=x[\/latex] by writing<\/p>\n<div id=\"fs-id1170572453591\" class=\"equation unnumbered\">[latex]f^{-1}(f(x))=f^{-1}(3x-4)=\\frac{1}{3}(3x-4)+\\frac{4}{3}=x-\\frac{4}{3}+\\frac{4}{3}=x[\/latex].<\/div>\n<p id=\"fs-id1170572546429\">Note that for [latex]f^{-1}(x)[\/latex] to be the inverse of [latex]f(x)[\/latex], both [latex]f^{-1}(f(x))=x[\/latex] and [latex]f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of the inside function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572478768\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572478772\" class=\"exercise\">\n<div id=\"fs-id1170572478774\" class=\"textbox\">\n<p id=\"fs-id1170572478776\">Find the inverse of the function [latex]f(x)=3x\/(x-2)[\/latex]. State the domain and range of the inverse function.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572479045\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572479045\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572479045\">[latex]f^{-1}(x)=\\frac{2x}{x-3}[\/latex]. The domain of [latex]f^{-1}[\/latex] is [latex]\\{x|x \\ne 3\\}[\/latex]. The range of [latex]f^{-1}[\/latex] is [latex]\\{y|y \\ne 2\\}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165042047502\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042050880\">Use the <a class=\"autogenerated-content\" href=\"#fs-id1170572552427\">(Note)<\/a> for finding inverse functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572363370\" class=\"bc-section section\">\n<h2>Graphing Inverse Functions<\/h2>\n<p id=\"fs-id1170572363375\">Let\u2019s consider the relationship between the graph of a function [latex]f[\/latex] and the graph of its inverse. Consider the graph of [latex]f[\/latex] shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_008\">(Figure)<\/a> and a point [latex](a,b)[\/latex] on the graph. Since [latex]b=f(a)[\/latex], then [latex]f^{-1}(b)=a[\/latex]. Therefore, when we graph [latex]f^{-1}[\/latex], the point [latex](b,a)[\/latex] is on the graph. As a result, the graph of [latex]f^{-1}[\/latex] is a reflection of the graph of [latex]f[\/latex] about the line [latex]y=x[\/latex].<\/p>\n<div id=\"CNX_Calc_Figure_01_04_008\" class=\"wp-caption aligncenter\">\n<div style=\"width: 720px\" 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\/11202543\/CNX_Calc_Figure_01_04_017.jpg\" alt=\"An image of two graphs. The first graph is of \u201cy = f(x)\u201d, which is a curved increasing function, that increases at a faster rate as x increases. The point (a, b) is on the graph of the function in the first quadrant. The second graph also graphs \u201cy = f(x)\u201d with the point (a, b), but also graphs the function \u201cy = f inverse (x)\u201d, an increasing curved function, that increases at a slower rate as x increases. This function includes the point (b, a). 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=\"710\" height=\"387\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong> (a) The graph of this function [latex]f[\/latex] shows point [latex](a,b)[\/latex] on the graph of [latex]f[\/latex]. (b) Since [latex](a,b)[\/latex] is on the graph of [latex]f[\/latex], the point [latex](b,a)[\/latex] is on the graph of [latex]f^{-1}[\/latex]. The graph of [latex]f^{-1}[\/latex] is a reflection of the graph of [latex]f[\/latex] about the line [latex]y=x[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572213184\" class=\"textbox examples\">\n<h3>Sketching Graphs of Inverse Functions<\/h3>\n<div id=\"fs-id1170572213190\" class=\"exercise\">\n<div id=\"fs-id1170572213192\" class=\"textbox\">\n<p id=\"fs-id1170572213194\">For the graph of [latex]f[\/latex] in the following image, sketch a graph of [latex]f^{-1}[\/latex] by sketching the line [latex]y=x[\/latex] and using symmetry. Identify the domain and range of [latex]f^{-1}[\/latex].<\/p>\n<p><span id=\"fs-id1170572222895\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202545\/CNX_Calc_Figure_01_04_009.jpg\" alt=\"An image of a graph. The x axis runs from -2 to 2 and the y axis runs from 0 to 2. The graph is of the function \u201cf(x) = square root of (x +2)\u201d, an increasing curved function. The function starts at the point (-2, 0). The x intercept is at (-2, 0) and the y intercept is at the approximate point (0, 1.4).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572222910\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572222910\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572222910\">Reflect the graph about the line [latex]y=x[\/latex]. The domain of [latex]f^{-1}[\/latex] is [latex][0,\\infty)[\/latex]. The range of [latex]f^{-1}[\/latex] is [latex][-2,\\infty)[\/latex]. By using the preceding strategy for finding inverse functions, we can verify that the inverse function is [latex]f^{-1}(x)=x^2-2[\/latex], as shown in the graph.<\/p>\n<p><span id=\"fs-id1170572480252\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202548\/CNX_Calc_Figure_01_04_010.jpg\" alt=\"An image of a graph. The x axis runs from -2 to 2 and the y axis runs from -2 to 2. The graph is of two functions. The first function is \u201cf(x) = square root of (x +2)\u201d, an increasing curved function. The function starts at the point (-2, 0). The x intercept is at (-2, 0) and the y intercept is at the approximate point (0, 1.4). The second function is \u201cf inverse (x) = (x squared) -2\u201d, an increasing curved function that starts at the point (0, -2). The x intercept is at the approximate point (1.4, 0) and the y intercept is at the point (0, -2). 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.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480264\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572480267\" class=\"exercise\">\n<div id=\"fs-id1170572480269\" class=\"textbox\">\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<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572481068\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572481068\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1170572481077\"><img 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.\" \/><\/span><\/div>\n<div id=\"fs-id1165041797872\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165041823836\">The graphs are symmetric about the line [latex]y=x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572481090\" class=\"bc-section section\">\n<h2>Restricting Domains<\/h2>\n<p id=\"fs-id1170572481095\">As we have seen, [latex]f(x)=x^2[\/latex] does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of [latex]f[\/latex] such that the function is one-to-one. This subset is called a restricted domain. By restricting the domain of [latex]f[\/latex], we can define a new function [latex]g[\/latex] such that the domain of [latex]g[\/latex] is the restricted domain of [latex]f[\/latex] and [latex]g(x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]g[\/latex]. Then we can define an inverse function for [latex]g[\/latex] on that domain. For example, since [latex]f(x)=x^2[\/latex] is one-to-one on the interval [latex][0,\\infty)[\/latex], we can define a new function [latex]g[\/latex] such that the domain of [latex]g[\/latex] is [latex][0,\\infty)[\/latex] and [latex]g(x)=x^2[\/latex] for all [latex]x[\/latex] in its domain. Since [latex]g[\/latex] is a one-to-one function, it has an inverse function, given by the formula [latex]g^{-1}(x)=\\sqrt{x}[\/latex]. On the other hand, the function [latex]f(x)=x^2[\/latex] is also one-to-one on the domain [latex](\u2212\\infty,0][\/latex]. Therefore, we could also define a new function [latex]h[\/latex] such that the domain of [latex]h[\/latex] is [latex](\u2212\\infty,0][\/latex] and [latex]h(x)=x^2[\/latex] for all [latex]x[\/latex] in the domain of [latex]h[\/latex]. Then [latex]h[\/latex] is a one-to-one function and must also have an inverse. Its inverse is given by the formula [latex]h^{-1}(x)=\u2212\\sqrt{x}[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_012\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_04_012\" class=\"wp-caption aligncenter\">\n<div style=\"width: 666px\" 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\/11202554\/CNX_Calc_Figure_01_04_012.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -2 to 5 and a y axis that runs from -2 to 5. The first graph is of two functions. The first function is \u201cg(x) = x squared\u201d, an increasing curved function that starts at the point (0, 0). This function increases at a faster rate for larger values of x. The second function is \u201cg inverse (x) = square root of x\u201d, an increasing curved function that starts at the point (0, 0). This function increases at a slower rate for larger values of x. The first function is \u201ch(x) = x squared\u201d, a decreasing curved function that ends at the point (0, 0). This function decreases at a slower rate for larger values of x. The second function is \u201ch inverse (x) = -(square root of x)\u201d, an increasing curved function that starts at the point (0, 0). This function decreases at a slower rate for larger values of x. 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=\"656\" height=\"353\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong> (a) For [latex]g(x)=x^2[\/latex] restricted to [latex][0,\\infty), \\, g^{-1}(x)=\\sqrt{x}[\/latex]. (b) For [latex]h(x)=x^2[\/latex] restricted to [latex](\u2212\\infty,0], \\, h^{-1}(x)=\u2212\\sqrt{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572551599\" class=\"textbox examples\">\n<h3>Restricting the Domain<\/h3>\n<div id=\"fs-id1170572551604\" class=\"exercise\">\n<div id=\"fs-id1170572551606\" class=\"textbox\">\n<p id=\"fs-id1170572551608\">Consider the function [latex]f(x)=(x+1)^2[\/latex].<\/p>\n<ol id=\"fs-id1170572551645\" style=\"list-style-type: lower-alpha\">\n<li>Sketch the graph of [latex]f[\/latex] and use the horizontal line test to show that [latex]f[\/latex] is not one-to-one.<\/li>\n<li>Show that [latex]f[\/latex] is one-to-one on the restricted domain [latex][-1,\\infty)[\/latex]. Determine the domain and range for the inverse of [latex]f[\/latex] on this restricted domain and find a formula for [latex]f^{-1}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572477860\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572477860\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572477860\" style=\"list-style-type: lower-alpha\">\n<li>The graph of [latex]f[\/latex] is the graph of [latex]y=x^2[\/latex] shifted left 1 unit. Since there exists a horizontal line intersecting the graph more than once, [latex]f[\/latex] is not one-to-one.<br \/>\n<span id=\"fs-id1170572477898\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202557\/CNX_Calc_Figure_01_04_013.jpg\" alt=\"An image of a graph. The x axis runs from -6 to 6 and the y axis runs from -2 to 10. The graph is of the function \u201cf(x) = (x+ 1) squared\u201d, which is a parabola. The function decreases until the point (-1, 0), where it begins it increases. The x intercept is at the point (-1, 0) and the y intercept is at the point (0, 1). There is also a horizontal dotted line plotted on the graph, which crosses through the function at two points.\" \/><\/span><\/li>\n<li>On the interval [latex][-1,\\infty), \\, f[\/latex] is one-to-one.<br \/>\n<span id=\"fs-id1170572236096\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202559\/CNX_Calc_Figure_01_04_014.jpg\" alt=\"An image of a graph. The x axis runs from -6 to 6 and the y axis runs from -2 to 10. The graph is of the function \u201cf(x) = (x+ 1) squared\u201d, on the interval &#091;1, infinity). The function starts from the point (-1, 0) and increases. The x intercept is at the point (-1, 0) and the y intercept is at the point (0, 1).\" \/><\/span><br \/>\nThe domain and range of [latex]f^{-1}[\/latex] are given by the range and domain of [latex]f[\/latex], respectively. Therefore, the domain of [latex]f^{-1}[\/latex] is [latex][0,\\infty)[\/latex] and the range of [latex]f^{-1}[\/latex] is [latex][-1,\\infty)[\/latex]. To find a formula for [latex]f^{-1}[\/latex], solve the equation [latex]y=(x+1)^2[\/latex] for [latex]x[\/latex]. If [latex]y=(x+1)^2[\/latex], then [latex]x=-1 \\pm \\sqrt{y}[\/latex]. Since we are restricting the domain to the interval where [latex]x \\ge -1[\/latex], we need [latex]\\pm \\sqrt{y} \\ge 0[\/latex]. Therefore, [latex]x=-1+\\sqrt{y}[\/latex]. Interchanging [latex]x[\/latex] and [latex]y[\/latex], we write [latex]y=-1+\\sqrt{x}[\/latex] and conclude that [latex]f^{-1}(x)=-1+\\sqrt{x}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572551765\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572551769\" class=\"exercise\">\n<div id=\"fs-id1170572551771\" class=\"textbox\">\n<p id=\"fs-id1170572551773\">Consider [latex]f(x)=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<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572141203\">Show Solution<\/span><\/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)=-1\/\\sqrt{x}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165041816987\" class=\"commentary\">\n<h4>Hint<\/h4>\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=1\/x^2[\/latex] for [latex]x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572141300\" class=\"bc-section section\">\n<h1>Inverse Trigonometric Functions<\/h1>\n<p id=\"fs-id1170572478796\">The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function (<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/trigonometric-functions\/#CNX_Calc_Figure_01_03_009\">(Figure)<\/a>). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex]. By doing so, we define the inverse sine function on the domain [latex][-1,1][\/latex] such that for any [latex]x[\/latex] in the interval [latex][-1,1][\/latex], the inverse sine function tells us which angle [latex]\\theta[\/latex] in the interval [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex] satisfies [latex]\\sin \\theta =x[\/latex]. Similarly, we can restrict the domains of the other trigonometric functions to define i<strong>nverse trigonometric functions<\/strong>, which are functions that tell us which angle in a certain interval has a specified trigonometric value.<\/p>\n<div id=\"fs-id1170572547395\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1170572547399\">The inverse sine function, denoted [latex]\\sin^{-1}[\/latex] or arcsin, and the inverse cosine function, denoted [latex]\\cos^{-1}[\/latex] or arccos, are defined on the domain [latex]D=\\{x|-1 \\le x \\le 1\\}[\/latex] as follows:<\/p>\n<div id=\"fs-id1170572547453\" class=\"equation\">[latex]\\begin{array}{c}\\sin^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\sin (y)=x \\, \\text{and} \\, -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2};\\hfill \\\\ \\cos^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\cos (y)=x \\, \\text{and} \\, 0 \\le y \\le \\pi \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572550974\">The inverse tangent function, denoted [latex]\\tan^{-1}[\/latex] or arctan, and inverse cotangent function, denoted [latex]\\cot^{-1}[\/latex] or arccot, are defined on the domain [latex]D=\\{x|-\\infty <x<\\infty \\}[\/latex] as follows:<\/p>\n<div id=\"fs-id1170572551028\" class=\"equation\">[latex]\\begin{array}{c}\\tan^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\tan (y)=x \\, \\text{and} \\, -\\frac{\\pi}{2}<y<\\frac{\\pi}{2};\\hfill \\\\ \\cot^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\cot (y)=x \\, \\text{and} \\, 0<y<\\pi \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572434001\">The inverse cosecant function, denoted [latex]\\csc^{-1}[\/latex] or arccsc, and inverse secant function, denoted [latex]\\sec^{-1}[\/latex] or arcsec, are defined on the domain [latex]D=\\{x| \\, |x| \\ge 1\\}[\/latex] as follows:<\/p>\n<div id=\"fs-id1170572548730\" class=\"equation\">[latex]\\begin{array}{c}\\csc^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\csc (y)=x \\, \\text{and} \\, -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2}, \\, y\\ne 0;\\hfill \\\\ \\sec^{-1}(x)=y \\,\\, \\text{if and only if} \\, \\sec (y)=x \\, \\text{and} \\, 0 \\le y \\le \\pi, \\, y \\ne \\pi\/2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1170572551163\">To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the domains defined earlier and reflect the graphs about the line [latex]y=x[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_015\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_04_015\" class=\"wp-caption aligncenter\">\n<div style=\"width: 861px\" 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\/11202603\/CNX_Calc_Figure_01_04_018.jpg\" alt=\"An image of six graphs. The first graph is of the function \u201cf(x) = sin inverse(x)\u201d, which is an increasing curve function. The function starts at the point (-1, -(pi\/2)) and increases until it ends at the point (1, (pi\/2)). The x intercept and y intercept are at the origin. The second graph is of the function \u201cf(x) = cos inverse (x)\u201d, which is a decreasing curved function. The function starts at the point (-1, pi) and decreases until it ends at the point (1, 0). The x intercept is at the point (1, 0). The y intercept is at the point (0, (pi\/2)). The third graph is of the function f(x) = tan inverse (x)\u201d, which is an increasing curve function. The function starts close to the horizontal line \u201cy = -(pi\/2)\u201d and increases until it comes close the \u201cy = (pi\/2)\u201d. The function never intersects either of these lines, it always stays between them - they are horizontal asymptotes. The x intercept and y intercept are both at the origin. The fourth graph is of the function \u201cf(x) = cot inverse (x)\u201d, which is a decreasing curved function. The function starts slightly below the horizontal line \u201cy = pi\u201d and decreases until it gets close the x axis. The function never intersects either of these lines, it always stays between them - they are horizontal asymptotes. The fifth graph is of the function \u201cf(x) = csc inverse (x)\u201d, a decreasing curved function. The function starts slightly below the x axis, then decreases until it hits a closed circle point at (-1, -(pi\/2)). The function then picks up again at the point (1, (pi\/2)), where is begins to decrease and approach the x axis, without ever touching the x axis. There is a horizontal asymptote at the x axis. The sixth graph is of the function \u201cf(x) = sec inverse (x)\u201d, an increasing curved function. The function starts slightly above the horizontal line \u201cy = (pi\/2)\u201d, then increases until it hits a closed circle point at (-1, pi). The function then picks up again at the point (1, 0), where is begins to increase and approach the horizontal line \u201cy = (pi\/2)\u201d, without ever touching the line. There is a horizontal asymptote at the \u201cy = (pi\/2)\u201d.\" width=\"851\" height=\"714\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong> The graph of each of the inverse trigonometric functions is a reflection about the line [latex]y=x[\/latex] of the corresponding restricted trigonometric function.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572366188\" class=\"textbox tryit media-2\">\n<p id=\"fs-id1170572366192\">Go to the <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_inversefun\">following site<\/a> for more comparisons of functions and their inverses.<\/p>\n<\/div>\n<p id=\"fs-id1170572366201\">When evaluating an inverse trigonometric function, the output is an angle. For example, to evaluate [latex]\\cos^{-1}(\\frac{1}{2})[\/latex], we need to find an angle [latex]\\theta[\/latex] such that [latex]\\cos \\theta =\\frac{1}{2}[\/latex]. Clearly, many angles have this property. However, given the definition of [latex]\\cos^{-1}[\/latex], we need the angle [latex]\\theta[\/latex] that not only solves this equation, but also lies in the interval [latex][0,\\pi][\/latex]. We conclude that [latex]\\cos^{-1}(\\frac{1}{2})=\\frac{\\pi}{3}[\/latex].<\/p>\n<p id=\"fs-id1170572472189\">We now consider a composition of a trigonometric function and its inverse. For example, consider the two expressions [latex]\\sin (\\sin^{-1}(\\frac{\\sqrt{2}}{2}))[\/latex] and [latex]\\sin^{-1}(\\sin(\\pi))[\/latex]. For the first one, we simplify as follows:<\/p>\n<div id=\"fs-id1170572472267\" class=\"equation unnumbered\">[latex]\\sin (\\sin^{-1}(\\frac{\\sqrt{2}}{2}))= \\sin (\\frac{\\pi}{4})=\\frac{\\sqrt{2}}{2}[\/latex].<\/div>\n<p>For the second one, we have<\/p>\n<div id=\"fs-id1170572548623\" class=\"equation unnumbered\">[latex]\\sin^{-1}( \\sin (\\pi))=\\sin^{-1}(0)=0[\/latex].<\/div>\n<p id=\"fs-id1170572548676\">The inverse function is supposed to \u201cundo\u201d the original function, so why isn\u2019t [latex]\\sin^{-1}(\\sin (\\pi))=\\pi[\/latex]? Recalling our definition of inverse functions, a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] satisfy the conditions [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in the domain of [latex]f^{-1}[\/latex] and [latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], so what happened here? The issue is that the inverse sine function, [latex]\\sin^{-1}[\/latex], is the inverse of the <em>restricted<\/em> sine function defined on the domain [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex]. Therefore, for [latex]x[\/latex] in the interval [latex][-\\frac{\\pi}{2},\\frac{\\pi}{2}][\/latex], it is true that [latex]\\sin^{-1}(\\sin x)=x[\/latex]. However, for values of [latex]x[\/latex] outside this interval, the equation does not hold, even though [latex]\\sin^{-1}(\\sin x)[\/latex] is defined for all real numbers [latex]x[\/latex].<\/p>\n<p id=\"fs-id1170572089863\">What about [latex]\\sin (\\sin^{-1}y)[\/latex]? Does that have a similar issue? The answer is <em>no<\/em>. Since the domain of [latex]\\sin^{-1}[\/latex] is the interval [latex][-1,1][\/latex], we conclude that [latex]\\sin (\\sin^{-1}y)=y[\/latex] if [latex]-1 \\le y \\le 1[\/latex] and the expression is not defined for other values of [latex]y[\/latex]. To summarize,<\/p>\n<div id=\"fs-id1170572550786\" class=\"equation unnumbered\">[latex]\\sin (\\sin^{-1}y)=y \\, \\text{if} \\, -1 \\le y \\le 1[\/latex]<\/div>\n<p id=\"fs-id1170572550837\">and<\/p>\n<div id=\"fs-id1170572550840\" class=\"equation unnumbered\">[latex]\\sin^{-1}( \\sin x)=x \\, \\text{if} \\, -\\frac{\\pi}{2} \\le x \\le \\frac{\\pi}{2}[\/latex].<\/div>\n<p id=\"fs-id1170572550900\">Similarly, for the cosine function,<\/p>\n<div id=\"fs-id1170572550903\" class=\"equation unnumbered\">[latex]\\cos (\\cos^{-1}y)=y \\, \\text{if} \\, -1 \\le y \\le 1[\/latex]<\/div>\n<p id=\"fs-id1170572549577\">and<\/p>\n<div id=\"fs-id1170572549580\" class=\"equation unnumbered\">[latex]\\cos^{-1}( \\cos x)=x \\, \\text{if} 0 \\le x \\le \\pi[\/latex].<\/div>\n<p id=\"fs-id1170572549631\">Similar properties hold for the other trigonometric functions and their inverses.<\/p>\n<div id=\"fs-id1170572549634\" class=\"textbox examples\">\n<h3>Evaluating Expressions Involving Inverse Trigonometric Functions<\/h3>\n<div id=\"fs-id1170572549640\" class=\"exercise\">\n<div id=\"fs-id1170572549642\" class=\"textbox\">\n<p id=\"fs-id1170572549644\">Evaluate each of the following expressions.<\/p>\n<ol id=\"fs-id1170572549647\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\sin^{-1}(-\\frac{\\sqrt{3}}{2})[\/latex]<\/li>\n<li>[latex]\\tan (\\tan^{-1}(-\\frac{1}{\\sqrt{3}}))[\/latex]<\/li>\n<li>[latex]\\cos^{-1}( \\cos (\\frac{5\\pi}{4}))[\/latex]<\/li>\n<li>[latex]\\sin^{-1}( \\cos (\\frac{2\\pi}{3}))[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572549361\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572549361\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572549361\" style=\"list-style-type: lower-alpha\">\n<li>Evaluating [latex]\\sin^{-1}(\u2212\\sqrt{3}\/2)[\/latex] is equivalent to finding the angle [latex]\\theta[\/latex] such that [latex]\\sin \\theta =\u2212\\sqrt{3}\/2[\/latex] and [latex]\u2212\\pi\/2 \\le \\theta \\le \\pi\/2[\/latex]. The angle [latex]\\theta =\u2212\\pi\/3[\/latex] satisfies these two conditions. Therefore, [latex]\\sin^{-1}(\u2212\\sqrt{3}\/2)=\u2212\\pi\/3[\/latex].<\/li>\n<li>First we use the fact that [latex]\\tan^{-1}(-1\/\\sqrt{3})=\u2212\\pi\/6[\/latex]. Then [latex]\\tan (\\pi\/6)=-1\/\\sqrt{3}[\/latex]. Therefore, [latex]\\tan (\\tan^{-1}(-1\/\\sqrt{3}))=-1\/\\sqrt{3}[\/latex].<\/li>\n<li>To evaluate [latex]\\cos^{-1}( \\cos (5\\pi\/4))[\/latex], first use the fact that [latex]\\cos (5\\pi\/4)=\u2212\\sqrt{2}\/2[\/latex]. Then we need to find the angle [latex]\\theta[\/latex] such that [latex]\\cos (\\theta )=\u2212\\sqrt{2}\/2[\/latex] and [latex]0 \\le \\theta \\le \\pi[\/latex]. Since [latex]3\\pi\/4[\/latex] satisfies both these conditions, we have [latex]\\cos (\\cos^{-1}(5\\pi\/4))= \\cos (\\cos^{-1}(\u2212\\sqrt{2}\/2))=3\\pi\/4[\/latex].<\/li>\n<li>Since [latex]\\cos (2\\pi\/3)=-1\/2[\/latex], we need to evaluate [latex]\\sin^{-1}(-1\/2)[\/latex]. That is, we need to find the angle [latex]\\theta[\/latex] such that [latex]\\sin (\\theta )=-1\/2[\/latex] and [latex]\u2212\\pi\/2 \\le \\theta \\le \\pi\/2[\/latex]. Since [latex]\u2212\\pi\/6[\/latex] satisfies both these conditions, we can conclude that [latex]\\sin^{-1}( \\cos (2\\pi\/3))=\\sin^{-1}(-1\/2)=\u2212\\pi\/6[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572169395\" class=\"textbox key-takeaways project\">\n<h3>The Maximum Value of a Function<\/h3>\n<p id=\"fs-id1170572169402\">In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don\u2019t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.<\/p>\n<p id=\"fs-id1170572169416\">This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable [latex]x[\/latex].<\/p>\n<ol id=\"fs-id1170572169426\">\n<li>Consider the graph in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_04_016\">(Figure)<\/a> of the function [latex]y= \\sin x + \\cos x[\/latex]. Describe its overall shape. Is it periodic? How do you know?\n<div id=\"CNX_Calc_Figure_01_04_016\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" 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\/11202607\/CNX_Calc_Figure_01_04_016.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of the function \u201cy = sin(x) + cos(x)\u201d, a curved wave function. The graph of the function decreases until it hits the approximate point (-(3pi\/4), -1.4), where it increases until the approximate point ((pi\/4), 1.4), where it begins to decrease again. The x intercepts shown on this graph of the function are at (-(5pi\/4), 0), (-(pi\/4), 0), and ((3pi\/4), 0). The y intercept is at (0, 1).\" width=\"325\" height=\"308\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6.<\/strong> The graph of [latex]y= \\sin x + \\cos x[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>Using a graphing calculator or other graphing device, estimate the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values of the maximum point for the graph (the first such point where [latex]x>0[\/latex]). It may be helpful to express the [latex]x[\/latex]-value as a multiple of [latex]\\pi[\/latex].<\/li>\n<li>Now consider other graphs of the form [latex]y=A \\sin x + B \\cos x[\/latex] for various values of [latex]A[\/latex] and [latex]B[\/latex]. Sketch the graph when [latex]A = 2[\/latex] and [latex]B = 1[\/latex], and find the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values for the maximum point. (Remember to express the [latex]x[\/latex]-value as a multiple of [latex]\\pi[\/latex], if possible.) Has it moved?<\/li>\n<li>Repeat for [latex]A = 1, \\, B = 2[\/latex]. Is there any relationship to what you found in part (2)?<\/li>\n<li>Complete the following table, adding a few choices of your own for [latex]A[\/latex] and [latex]B[\/latex]:<br \/>\n<table id=\"fs-id1170572554057\" class=\"unnumbered\" summary=\"A table containing 8 columns and 9 rows is shown. The first column is labeled \u201cA\u201d and contains the values \u201c0,1,1,1,2,2,3, and 4.\u201d The second column is labeled \u201cB\u201d and contains the values \u201c1,0,1,2,1,2,4, and 3.\u201d The third column is labeled \u201cx\u201d and has no values for any of the rows. The fourth column is labeled \u201cy\u201d and contains no values for any of the rows. The fifth column is separated from the fourth column by a gutter, is labeled \u201cA\u201d and contains the values \u201cthe square root of 3,1,12, and 5.\u201d The sixth column is labeled \u201cB\u201d and contains the values 1, the square root of 3,5, and 12.\u201d The seventh column is labeled \u201cx\u201d and contains no values for any of the rows. The eighth column is labeled \u201cy\u201d and contains no values for any of the rows.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]A[\/latex]<\/th>\n<th>[latex]B[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th><\/th>\n<th>[latex]A[\/latex]<\/th>\n<th>[latex]B[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>0<\/td>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<td rowspan=\"8\"><\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>0<\/td>\n<td><\/td>\n<td><\/td>\n<td>1<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<td>12<\/td>\n<td>5<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<td>5<\/td>\n<td>12<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>1<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>2<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3<\/td>\n<td>4<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>4<\/td>\n<td>3<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Try to figure out the formula for the [latex]y[\/latex]-values.<\/li>\n<li>The formula for the [latex]x[\/latex]-values is a little harder. The most helpful points from the table are [latex](1,1), \\, (1,\\sqrt{3}), \\, (\\sqrt{3},1)[\/latex]. (<em>Hint<\/em>: <em>Consider inverse trigonometric functions.)<\/em><\/li>\n<li>If you found formulas for parts (5) and (6), show that they work together. That is, substitute the [latex]x[\/latex]-value formula you found into [latex]y=A \\sin x + B \\cos x[\/latex] and simplify it to arrive at the [latex]y[\/latex]-value formula you found.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572470426\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1170572470433\">\n<li>For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.<\/li>\n<li>If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.<\/li>\n<li>For a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f^{-1}[\/latex] and [latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\n<li>Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.<\/li>\n<li>The graph of a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] are symmetric about the line [latex]y=x[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572453118\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<ul id=\"fs-id1170572453126\">\n<li><strong>Inverse functions<\/strong><br \/>\n[latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in [latex]D[\/latex], and [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in [latex]R[\/latex].[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572547960\" class=\"textbox exercises\">\n<p id=\"fs-id1170572547964\">For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.<\/p>\n<div id=\"fs-id1170572547969\" class=\"exercise\">\n<div id=\"fs-id1170572547971\" class=\"textbox\"><span id=\"fs-id1170572547973\"><strong>1.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202609\/CNX_Calc_Figure_01_04_201.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that decreases in a straight in until the origin, where it begins to increase in a straight line. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572547990\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572547990\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572547990\">Not one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572547995\" class=\"exercise\">\n<div id=\"fs-id1170572547997\" class=\"textbox\"><span id=\"fs-id1170572547999\"><strong>2.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202612\/CNX_Calc_Figure_01_04_202.jpg\" alt=\"An image of a graph. The x axis runs from 0 to 7 and the y axis runs from -4 to 4. The graph is of a function that is always increasing. There is an approximate x intercept at the point (1, 0) and no y intercept shown.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1170572548022\" class=\"exercise\">\n<div id=\"fs-id1170572548024\" class=\"textbox\"><span id=\"fs-id1170572548026\"><strong>3.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202614\/CNX_Calc_Figure_01_04_203.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that resembles a semi-circle, the top half of a circle. The function starts at the point (-3, 0) and increases until the point (0, 3), where it begins decreasing until it ends at the point (3, 0). The x intercepts are at (-3, 0) and (3, 0). The y intercept is at (0, 3).\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572548044\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572548044\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572548044\">Not one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572548050\" class=\"exercise\">\n<div id=\"fs-id1170572548052\" class=\"textbox\"><span id=\"fs-id1170572548054\"><strong>4.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202617\/CNX_Calc_Figure_01_04_204.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function. The function increases until it hits the origin, then decreases until it hits the point (2, -4), where it begins to increase again. There are x intercepts at the origin and the point (3, 0). The y intercept is at the origin.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1170572548077\" class=\"exercise\">\n<div id=\"fs-id1170572548079\" class=\"textbox\"><span id=\"fs-id1170572548081\"><strong>5.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202620\/CNX_Calc_Figure_01_04_205.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function that is always increasing. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572548098\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572548098\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572548098\">One-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452097\" class=\"exercise\">\n<div id=\"fs-id1170572452099\" class=\"textbox\"><span id=\"fs-id1170572452101\"><strong>6.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202622\/CNX_Calc_Figure_01_04_206.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 7 and the y axis runs from -4 to 4. The graph is of a function that increases in a straight line until the approximate point (, 3). After this point, the function becomes a horizontal straight line. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170572452124\">For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.<\/p>\n<div id=\"fs-id1170572452128\" class=\"exercise\">\n<div id=\"fs-id1170572452130\" class=\"textbox\">\n<p id=\"fs-id1170572452132\"><strong>7.\u00a0<\/strong>[latex]f(x)=x^2-4, \\, x \\ge 0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572452170\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572452170\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572452170\">a. [latex]f^{-1}(x)=\\sqrt{x+4}[\/latex] b. Domain: [latex]x \\ge -4[\/latex], Range: [latex]y \\ge 0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452229\" class=\"exercise\">\n<div id=\"fs-id1170572452231\" class=\"textbox\">\n<p id=\"fs-id1170572452233\"><strong>8.\u00a0<\/strong>[latex]f(x)=\\sqrt[3]{x-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572176863\" class=\"exercise\">\n<div id=\"fs-id1170572176865\" class=\"textbox\">\n<p id=\"fs-id1170572176867\"><strong>9.\u00a0<\/strong>[latex]f(x)=x^3+1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572176896\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572176896\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572176896\">a. [latex]f^{-1}(x)=\\sqrt[3]{x-1}[\/latex] b. Domain: all real numbers, Range: all real numbers<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572176931\" class=\"exercise\">\n<div id=\"fs-id1170572176934\" class=\"textbox\">\n<p id=\"fs-id1170572176936\"><strong>10.\u00a0<\/strong>[latex]f(x)=(x-1)^2, \\, x \\le 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572549738\" class=\"exercise\">\n<div id=\"fs-id1170572549740\" class=\"textbox\">\n<p id=\"fs-id1170572549742\"><strong>11.\u00a0<\/strong>[latex]f(x)=\\sqrt{x-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572549770\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572549770\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572549770\">a. [latex]f^{-1}(x)=x^2+1[\/latex], b. Domain: [latex]x \\ge 0[\/latex], Range: [latex]y \\ge 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572549826\" class=\"exercise\">\n<div id=\"fs-id1170572549828\" class=\"textbox\">\n<p id=\"fs-id1170572549831\"><strong>12.\u00a0<\/strong>[latex]f(x)=\\frac{1}{x+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572451519\">For the following exercises, use the graph of [latex]f[\/latex] to sketch the graph of its inverse function.<\/p>\n<div id=\"fs-id1170572451526\" class=\"exercise\">\n<div id=\"fs-id1170572451528\" class=\"textbox\"><span id=\"fs-id1170572451534\"><strong>13.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202625\/CNX_Calc_Figure_01_04_207.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of an increasing straight line function labeled \u201cf\u201d that is always increasing. The x intercept is at (-2, 0) and y intercept are both at (0, 1).\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572451548\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572451548\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1170572451558\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202627\/CNX_Calc_Figure_01_04_208.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of two functions. The first function is an increasing straight line function labeled \u201cf\u201d. The x intercept is at (-2, 0) and y intercept are both at (0, 1). The second function is of an increasing straight line function labeled \u201cf inverse\u201d. The x intercept is at the point (1, 0) and the y intercept is at the point (0, -2).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572451569\" class=\"exercise\">\n<div id=\"fs-id1170572451571\" class=\"textbox\"><span id=\"fs-id1170572451577\"><strong>14.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202630\/CNX_Calc_Figure_01_04_209.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved decreasing function labeled \u201cf\u201d. As the function decreases, it gets approaches the x axis but never touches it. The function does not have an x intercept and the y intercept is (0, 1).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1170572452480\" class=\"exercise\">\n<div id=\"fs-id1170572452482\" class=\"textbox\"><span id=\"fs-id1170572452491\"><strong>15.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202633\/CNX_Calc_Figure_01_04_211.jpg\" alt=\"An image of a graph. The x axis runs from -8 to 8 and the y axis runs from -8 to 8. The graph is of an increasing straight line function labeled \u201cf\u201d. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1).\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572452503\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572452503\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1170572452513\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202636\/CNX_Calc_Figure_01_04_212.jpg\" alt=\"An image of a graph. The x axis runs from 0 to 8 and the y axis runs from 0 to 8. The graph is of two function. The first function is an increasing straight line function labeled \u201cf\u201d. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1). The second function is an increasing straight line function labeled \u201cf inverse\u201d. The function starts at the point (1, 0) and increases in straight line until the point (6, 4). After this point, the function continues to increase, but at a faster rate than before, as it approaches the point (8, 8). The function does not have an y intercept and the x intercept is (1, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452528\" class=\"exercise\">\n<div id=\"fs-id1170572452530\" class=\"textbox\"><span id=\"fs-id1170572452536\"><strong>16.<br \/>\n<\/strong><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202639\/CNX_Calc_Figure_01_04_213.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a decreasing curved function labeled \u201cf\u201d, which ends at the origin, which is both the x intercept and y intercept. Another point on the function is (-4, 2).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170572452571\">For the following exercises, use composition to determine which pairs of functions are inverses.<\/p>\n<div id=\"fs-id1170572452575\" class=\"exercise\">\n<div id=\"fs-id1170572452577\" class=\"textbox\">\n<p id=\"fs-id1170572452579\"><strong>17.\u00a0<\/strong>[latex]f(x)=8x, \\, g(x)=\\frac{x}{8}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572452622\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572452622\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572452622\">These are inverses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572452632\"><strong>18.\u00a0<\/strong>[latex]f(x)=8x+3, \\, g(x)=\\frac{x-3}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572449207\" class=\"exercise\">\n<div id=\"fs-id1170572449209\" class=\"textbox\">\n<p id=\"fs-id1170572449211\"><strong>19.\u00a0<\/strong>[latex]f(x)=5x-7, \\, g(x)=\\frac{x+5}{7}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572449263\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572449263\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572449263\">These are not inverses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572449269\" class=\"exercise\">\n<div id=\"fs-id1170572449271\" class=\"textbox\">\n<p id=\"fs-id1170572449273\"><strong>20.\u00a0<\/strong>[latex]f(x)=\\frac{2}{3}x+2, \\, g(x)=\\frac{3}{2}x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572548356\" class=\"exercise\">\n<div id=\"fs-id1170572548358\" class=\"textbox\">\n<p id=\"fs-id1170572548360\"><strong>21.\u00a0<\/strong>[latex]f(x)=\\frac{1}{x-1}, \\, x \\ne 1, \\, g(x)=\\frac{1}{x}+1, \\, x \\ne 0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572548430\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572548430\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572548430\">These are inverses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572548436\" class=\"exercise\">\n<div id=\"fs-id1170572548438\" class=\"textbox\">\n<p id=\"fs-id1170572548440\"><strong>22. <\/strong>[latex]f(x)=x^3+1, \\, g(x)=(x-1)^{1\/3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572548511\" class=\"exercise\">\n<div id=\"fs-id1170572229236\" class=\"textbox\">\n<p id=\"fs-id1170572229238\"><strong>23.\u00a0<\/strong>[latex]f(x)=x^2+2x+1, \\, x \\ge -1,\\, g(x)=-1+\\sqrt{x}, \\, x \\ge 0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572229326\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572229326\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572229326\">These are inverses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572229332\" class=\"exercise\">\n<div id=\"fs-id1170572229334\" class=\"textbox\">\n<p id=\"fs-id1170572229336\"><strong>24.\u00a0<\/strong>[latex]f(x)=\\sqrt{4-x^2}, \\, 0 \\le x \\le 2, \\, g(x)=\\sqrt{4-x^2}, \\, 0 \\le x \\le 2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572451285\">For the following exercises, evaluate the functions. Give the exact value.<\/p>\n<div id=\"fs-id1170572451288\" class=\"exercise\">\n<div id=\"fs-id1170572451291\" class=\"textbox\">\n<p id=\"fs-id1170572451293\"><strong>25.\u00a0<\/strong>[latex]\\tan^{-1}(\\frac{\\sqrt{3}}{3})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572451322\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572451322\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572451322\">[latex]\\frac{\\pi}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572451334\" class=\"exercise\">\n<div id=\"fs-id1170572451336\" class=\"textbox\">\n<p><strong>26.\u00a0<\/strong>[latex]\\cos^{-1}(-\\frac{\\sqrt{2}}{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572451384\" class=\"exercise\">\n<div id=\"fs-id1170572451386\" class=\"textbox\">\n<p id=\"fs-id1170572451388\"><strong>27.\u00a0<\/strong>[latex]\\cot^{-1}(1)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572451411\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572451411\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572451411\">[latex]\\frac{\\pi}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572451423\" class=\"exercise\">\n<div id=\"fs-id1170572451425\" class=\"textbox\">\n<p id=\"fs-id1170572451427\"><strong>28.\u00a0<\/strong>[latex]\\sin^{-1}(-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572142146\" class=\"exercise\">\n<div id=\"fs-id1170572142148\" class=\"textbox\">\n<p id=\"fs-id1170572142150\"><strong>29.\u00a0<\/strong>[latex]\\cos^{-1}(\\frac{\\sqrt{3}}{2})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572142179\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572142179\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572142179\">[latex]\\frac{\\pi}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572142191\" class=\"exercise\">\n<div id=\"fs-id1170572142193\" class=\"textbox\">\n<p id=\"fs-id1170572142195\"><strong>30.\u00a0<\/strong>[latex]\\cos (\\tan^{-1}(\\sqrt{3}))[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572142240\" class=\"exercise\">\n<div id=\"fs-id1170572142242\" class=\"textbox\">\n<p id=\"fs-id1170572142245\"><strong>31.\u00a0<\/strong>[latex]\\sin (\\cos^{-1}(\\frac{\\sqrt{2}}{2}))[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q461959\">Show Answer<\/span><\/p>\n<div id=\"q461959\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572142296\" class=\"exercise\">\n<div id=\"fs-id1170572548960\" class=\"textbox\">\n<p id=\"fs-id1170572548962\"><strong>32.\u00a0<\/strong>[latex]\\sin^{-1}( \\sin (\\frac{\\pi}{3}))[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572549009\" class=\"exercise\">\n<div id=\"fs-id1170572549011\" class=\"textbox\">\n<p id=\"fs-id1170572549013\"><strong>33.\u00a0<\/strong>[latex]\\tan^{-1}( \\tan (-\\frac{\\pi}{6}))[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572549051\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572549051\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572549051\">[latex]-\\frac{\\pi}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572549065\" class=\"exercise\">\n<div id=\"fs-id1170572549067\" class=\"textbox\">\n<p id=\"fs-id1170572549069\"><strong>34.\u00a0<\/strong>The function [latex]C=T(F)=(5\/9)(F-32)[\/latex] converts degrees Fahrenheit to degrees Celsius.<\/p>\n<ol id=\"fs-id1170572549118\" style=\"list-style-type: lower-alpha\">\n<li>Find the inverse function [latex]F=T^{-1}(C)[\/latex]<\/li>\n<li>What is the inverse function used for?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572451744\" class=\"exercise\">\n<div id=\"fs-id1170572451746\" class=\"textbox\">\n<p id=\"fs-id1170572451748\"><strong>35. [T]<\/strong> The velocity [latex]V[\/latex] (in centimeters per second) of blood in an artery at a distance [latex]x[\/latex] cm from the center of the artery can be modeled by the function [latex]V=f(x)=500(0.04-x^2)[\/latex] for [latex]0 \\le x \\le 0.2[\/latex].<\/p>\n<ol id=\"fs-id1170572451823\" style=\"list-style-type: lower-alpha\">\n<li>Find [latex]x=f^{-1}(V)[\/latex].<\/li>\n<li>Interpret what the inverse function is used for.<\/li>\n<li>Find the distance from the center of an artery with a velocity of 15 cm\/sec, 10 cm\/sec, and 5 cm\/sec.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572547753\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572547753\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572547753\">a. [latex]x=f^{-1}(V)=\\sqrt{0.04-\\frac{V}{500}}[\/latex] b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity [latex]V[\/latex]. c. 0.1 cm; 0.14 cm; 0.17 cm<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572547801\" class=\"exercise\">\n<div id=\"fs-id1170572547803\" class=\"textbox\">\n<p id=\"fs-id1170572547805\"><strong>36.\u00a0<\/strong>A function that converts dress sizes in the United States to those in Europe is given by [latex]D(x)=2x+24[\/latex].<\/p>\n<ol id=\"fs-id1170572547832\" style=\"list-style-type: lower-alpha\">\n<li>Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States.<\/li>\n<li>Find the function that converts European dress sizes to U.S. dress sizes.<\/li>\n<li>Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572547878\" class=\"exercise\">\n<div id=\"fs-id1170572547880\" class=\"textbox\">\n<p id=\"fs-id1170572547882\"><strong>37. [T]<\/strong> The cost to remove a toxin from a lake is modeled by the function<\/p>\n<p id=\"fs-id1170572547889\">[latex]C(p)=75p\/(85-p)[\/latex], where [latex]C[\/latex] is the cost (in thousands of dollars) and [latex]p[\/latex] is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.<\/p>\n<ol id=\"fs-id1170572542869\" style=\"list-style-type: lower-alpha\">\n<li>Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.<\/li>\n<li>Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572542887\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572542887\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572542887\">a. $31,250, $66,667, $107,143 b. [latex](p=\\frac{85C}{C+75})[\/latex] c. 34 ppb<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572542921\" class=\"exercise\">\n<div id=\"fs-id1170572542923\" class=\"textbox\">\n<p id=\"fs-id1170572542925\"><strong>38. [T]<\/strong> A race car is accelerating at a velocity given by<\/p>\n<p id=\"fs-id1170572542932\">[latex]v(t)=\\frac{25}{4}t+54[\/latex],<\/p>\n<p id=\"fs-id1170572542966\">where [latex]v[\/latex] is the velocity (in feet per second) at time [latex]t[\/latex].<\/p>\n<ol id=\"fs-id1170572542980\" style=\"list-style-type: lower-alpha\">\n<li>Find the velocity of the car at 10 sec.<\/li>\n<li>Find the inverse function.<\/li>\n<li>Use part b. to determine how long it takes for the car to reach a speed of 150 ft\/sec.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572543027\" class=\"exercise\">\n<div id=\"fs-id1170572455080\" class=\"textbox\">\n<p id=\"fs-id1170572455082\"><strong>39. [T]<\/strong> An airplane\u2019s Mach number [latex]M[\/latex] is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by [latex]\\mu =2\\sin^{-1}(\\frac{1}{M})[\/latex].<\/p>\n<p id=\"fs-id1170572455127\">Find the Mach angle (to the nearest degree) for the following Mach numbers.<\/p>\n<p><span id=\"fs-id1170572455137\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202641\/CNX_Calc_Figure_01_04_215.jpg\" alt=\"An image of a birds eye view of an airplane. Directly in front of the airplane is a sideways \u201cV\u201d shape, with the airplane flying directly into the opening of the \u201cV\u201d shape. The \u201cV\u201d shape is labeled \u201cmach wave\u201d. There are two arrows with labels. The first arrow points from the nose of the airplane to the corner of the \u201cV\u201d shape. This arrow has the label \u201cvelocity = v\u201d. The second arrow points diagonally from the nose of the airplane to the edge of the upper portion of the \u201cV\u201d shape. This arrow has the label \u201cspeed of sound = a\u201d. Between these two arrows is an angle labeled \u201cMach angle\u201d. There is also text in the image that reads \u201cmach = M &gt; 1.0\u201d.\" \/><\/span><\/p>\n<ol id=\"fs-id1170572455148\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\mu =1.4[\/latex]<\/li>\n<li>[latex]\\mu =2.8[\/latex]<\/li>\n<li>[latex]\\mu =4.3[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572455191\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572455191\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572455191\">a. [latex]~92^{\\circ}[\/latex] b. [latex]~42^{\\circ}[\/latex] c. [latex]~27^{\\circ}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572455224\" class=\"exercise\">\n<div id=\"fs-id1170572455226\" class=\"textbox\">\n<p id=\"fs-id1170572455228\"><strong>40. [T]<\/strong> Using [latex]\\mu =2\\sin^{-1}(\\frac{1}{M})[\/latex], find the Mach number [latex]M[\/latex] for the following angles.<\/p>\n<ol id=\"fs-id1170572551412\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\mu =\\frac{\\pi}{6}[\/latex]<\/li>\n<li>[latex]\\mu =\\frac{2\\pi}{7}[\/latex]<\/li>\n<li>[latex]\\mu =\\frac{3\\pi}{8}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572551494\" class=\"exercise\">\n<div id=\"fs-id1170572551497\" class=\"textbox\">\n<p id=\"fs-id1170572551499\"><strong>41. [T]<\/strong> The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function<\/p>\n<p id=\"fs-id1170572551507\">[latex]T(x)=5+18 \\sin[\\frac{\\pi}{6}(x-4.6)][\/latex],<\/p>\n<p id=\"fs-id1170572551558\">where [latex]x[\/latex] is time in months and [latex]x=1.00[\/latex] corresponds to January 1. Determine the month and day when the temperature is [latex]21^{\\circ}[\/latex] C.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572545094\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572545094\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572545094\">[latex]x \\approx 6.69,8.51[\/latex]; so, the temperature occurs on June 21 and August 15<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572545115\" class=\"exercise\">\n<div id=\"fs-id1170572545117\" class=\"textbox\">\n<p id=\"fs-id1170572545119\"><strong>42. [T]<\/strong> The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function<\/p>\n<p id=\"fs-id1170572545127\">[latex]D(t)=5 \\sin (\\frac{\\pi}{6}t-\\frac{7\\pi}{6})+8[\/latex],<\/p>\n<p id=\"fs-id1170572545179\">where [latex]t[\/latex] is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572545202\" class=\"exercise\">\n<div id=\"fs-id1170572545204\" class=\"textbox\">\n<p id=\"fs-id1170572545207\"><strong>43. [T]<\/strong> An object moving in simple harmonic motion is modeled by the function<\/p>\n<p id=\"fs-id1170572545214\">[latex]s(t)=-6 \\cos (\\frac{\\pi t}{2})[\/latex],<\/p>\n<p id=\"fs-id1170572545252\">where [latex]s[\/latex] is measured in inches and [latex]t[\/latex] is measured in seconds. Determine the first time when the distance moved is 4.5 ft.<\/p>\n<\/div>\n<div id=\"fs-id1170572545265\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572545265\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572545265\" class=\"hidden-answer\" style=\"display: none\">[latex]~1.5 \\sec[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572169073\" class=\"exercise\">\n<div id=\"fs-id1170572169076\" class=\"textbox\">\n<p id=\"fs-id1170572169078\"><strong>44. [T]<\/strong> A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle [latex]\\theta[\/latex] can be modeled by the function<\/p>\n<p id=\"fs-id1170572169091\">[latex]\\theta =\\tan^{-1}\\frac{5.5}{x}-\\tan^{-1}\\frac{2.5}{x}[\/latex],<\/p>\n<p id=\"fs-id1170572169132\">where [latex]x[\/latex] is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572169169\" class=\"exercise\">\n<div id=\"fs-id1170572169171\" class=\"textbox\">\n<p id=\"fs-id1170572169173\"><strong>45. [T]<\/strong> Use a calculator to evaluate [latex]\\tan^{-1}( \\tan (2.1))[\/latex] and [latex]\\cos^{-1}( \\cos (2.1))[\/latex]. Explain the results of each.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572243719\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572243719\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572243719\">[latex]\\tan^{-1}( \\tan (2.1))\\approx -1.0416[\/latex]; the expression does not equal 2.1 since [latex]2.1>1.57=\\frac{\\pi}{2}[\/latex]\u2014in other words, it is not in the restricted domain of [latex]\\tan x[\/latex].\u00a0 [latex]\\cos^{-1}( \\cos (2.1))=2.1[\/latex], since 2.1 is in the restricted domain of [latex]\\cos x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572243833\" class=\"exercise\">\n<div id=\"fs-id1170572243835\" class=\"textbox\">\n<p><strong>46. [T]<\/strong> Use a calculator to evaluate [latex]\\sin (\\sin^{-1}(-2))[\/latex] and [latex]\\tan (\\tan^{-1}(-2))[\/latex]. Explain the results of each.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572229168\" class=\"definition\">\n<dt>horizontal line test<\/dt>\n<dd id=\"fs-id1170572229174\">a function [latex]f[\/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[\/latex], at most, once<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572229190\" class=\"definition\">\n<dt>inverse function<\/dt>\n<dd id=\"fs-id1170572229195\">for a function [latex]f[\/latex], the inverse function [latex]f^{-1}[\/latex] satisfies [latex]f^{-1}(y)=x[\/latex] if [latex]f(x)=y[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572482608\" class=\"definition\">\n<dt>inverse trigonometric functions<\/dt>\n<dd id=\"fs-id1170572482614\">the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572482619\" class=\"definition\">\n<dt>one-to-one function<\/dt>\n<dd id=\"fs-id1170572482624\">a function [latex]f[\/latex] is one-to-one if [latex]f(x_1) \\ne f(x_2)[\/latex] if [latex]x_1 \\ne x_2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572482683\" class=\"definition\">\n<dt>restricted domain<\/dt>\n<dd id=\"fs-id1170572482689\">a subset of the domain of a function [latex]f[\/latex]<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":311,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1559","chapter","type-chapter","status-publish","hentry"],"part":1448,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1559","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1559\/revisions"}],"predecessor-version":[{"id":2529,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1559\/revisions\/2529"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1448"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1559\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1559"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1559"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1559"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}