Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
1/10/2014 MTH 125 1.5 Inverse Functions •Recall that a function can be represented by a set of ordered pairs. For instance, the function f (x) = x + 3 from A = {1, 2, 3, 4} to B = {4, 5, 6, 7} can be written as • f : {(1, 4), (2, 5), (3, 6), (4, 7)}. •By interchanging the first and second coordinates of each ordered pair, you can form the inverse function of f. This function is denoted by f –1. It is a function from B to A, and can be written as • f –1: {(4, 1), (5, 2), (6, 3), (7, 4)}. 1.5 Copyright © Cengage Learning. All rights reserved. Inverse Functions •Note that the domain of f is equal to the range of f –1, and vice versa, as shown in Figure 1.35. Domain of f = range of f –1 Domain of f –1 = range of f Figure 1.35 1 1/10/2014 Inverse Functions •That is, when you form the composition of f with f –1 or the composition of f –1 with f, you obtain the identity function. • f(f –1(x)) = x and f –1(f(x)) = x Inverse Functions •Note: •Although the notation used to denote an inverse function resembles exponential notation, it is a different use of –1 as a superscript. •That is, in general, f –1(x) ≠ 1/f(x). 2 1/10/2014 Inverse Functions •Here are some important observations about inverse functions. •1. If g is the inverse function of f, then f is the inverse function of g. •2. The domain of f –1 is equal to the range of f, and the range of f –1 is equal to the domain of f. •3. A function need not have an inverse function, but if it does, the inverse function is unique. •You can think of f –1 as undoing what has been done by f. Example 1 – Verifying Inverse Functions •Show that the functions are inverse functions of each other. •f(x) = 2x3 – 1 and g(x) = •Solution: •Because the domains and ranges of both f and g consist of all real numbers, you can conclude that both composite functions exist for all x. •The composition of f with g is given by 3 1/10/2014 Example 1 – Solution cont’d • =x+1–1 • = x. •The composition of g with f is given by Example 1 – Solution • cont’d = x. •Because f(g(x)) = x and g(f(x)) = x, you can conclude that f and g are inverse functions of each other (see Figure 1.36). f and g are inverse functions of each other. Figure 1.36 4 1/10/2014 Inverse Functions •In Figure 1.36, the graphs of f and g = f –1 appear to be mirror images of each other with respect to the line y = x. •The graph of f –1 is a reflection of the graph of f in the line y = x. •This idea is generalized as follows. Show analytically that f and g are inverse functions f ( x ) = 3 − 4 x, g ( x) = 3− x 4 5 1/10/2014 Existence of an Inverse Function •Not every function has an inverse, and the Reflective Property of Inverse Functions suggests a graphical test for those that do—the Horizontal Line Test for an inverse function. •This test states that a function f has an inverse function if and only if every horizontal line intersects the graph of f at most If a horizontal line intersects the graph of f twice, then f is not one-to-one. once (see Figure 1.38). Figure 1.38 Use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. 6 1/10/2014 Find the inverse function of f, graph f and f −1 on the same set of axes, describe the relationship between the graphs, and state the domains and −1 ranges of f and f . f ( x) = x 3 − 1 Inverse Trigonometric Functions The six trig functions do not pass the horizontal line test, so they do not have inverse functions. To find the inverse of a trig function, you must first restrict its domain. 7 1/10/2014 The sine function is one-to-one on the interval [ -π/2, π/2] y = sin x, -π/2 ≤ x ≤ π/2 Domain: [-π/2 , π/2] Range: [-1, 1] y = arcsin x, -1≤ x ≤ 1 Domain: [-1, 1] Range: [-π/2 , π/2] The cosine function is one-to-one on the interval [0,π]. The tangent function is one-to-one on the interval (-π/2, π/2 ) Definition of Inverse Trigonometric Function chart on page 41 8 1/10/2014 Evaluating Inverse Trig Functions Evaluate each of the following. arcsin 1/2 arccos (– ½) arctan 3 arcsin (0.8) Properties of Inverse Trig Functions 1. sin(arcsin x) = x and arcsin(siny) = y if -1 ≤ x ≤ 1 and –π/2 ≤ y ≤ π/2 2. tan(arctanx) = x and arctan(tany) = y if –π/2 < y < π/2 3. sec(arcsec x) = x and arcsec(sec y) = y if |x| ≥ 1 and 0≤ y < π/2 or π/2< y≤ π 9 1/10/2014 Using Right Triangles Evaluate without using a calculator: cos(arcsin 5/13) sin θ = opp/hyp 5 13 Find the adjacent side using the Pythagorean Theorem. θ cos θ = adj/hyp 10