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4.1 Inverse Functions Copyright © 2007 Pearson Education, Inc. Slide 5-1 5.1 One-to-One Functions A function f is a one-to-one function if, for elements a and b from the domain of f, ab • implies f (a) f (b). Only functions that are one-to-one have inverses. Copyright © 2007 Pearson Education, Inc. Slide 5-2 5.1 One-to-One Functions Example Decide whether the function is one-to-one. (a) f ( x) 4 x 12 (b) f ( x) 25 x 2 Solution (a) For this function, two different x-values produce two different y-values. Suppose that a b, then 4a 4b and 4a 12 4b 12. Since f (a) f (b), f is one - to - one. (b) If we choose a = 3 and b = –3, then 3 –3, but f (3) 25 32 4 and f (3) 25 (3) 2 4, so f (3) f (3), therefore f is not one - to - one. Copyright © 2007 Pearson Education, Inc. Slide 5-3 5.1 The Horizontal Line Test If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions. (a) (b) Not one-to-one Copyright © 2007 Pearson Education, Inc. One-to-one Slide 5-4 5.1 Inverse Functions Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if ( f g )( x) x for every x in the domain of g , and ( g f )( x) x for every x in the domain of f . Example Show that f ( x) x3 1 and g ( x) 3 x 1 are inverse functions of each other. ( f g )( x) f [ g ( x)] x 1 1 x 1 1 x 3 3 ( g f )( x) g[ f ( x)] 3 x 3 1 1 3 x 3 x Copyright © 2007 Pearson Education, Inc. Slide 5-5 5.1 Finding an Equation for the Inverse Function • Notation for the inverse function f -1 is read “f-inverse” Finding the Equation of the Inverse of y = f(x) 1. Interchange x and y. 2. Solve for y. 3. Replace y with f -1(x). Any restrictions on x and y should be considered. Copyright © 2007 Pearson Education, Inc. Slide 5-6 5.1 Example of Finding f -1(x) Example Find the inverse, if it exists, of 4x 6 f ( x) . 5 4x 6 Write f (x) = y. y Solution 5 4y 6 Interchange x and y. x 5 Solve for y. 5x 4 y 6 5x 6 y 4 5x 6 1 Replace y with f -1(x). f ( x) 4 Copyright © 2007 Pearson Education, Inc. Slide 5-7 5.1 The Graph of f -1(x) • f and f -1(x) are inverse functions, and f (a) = b for real numbers a and b. Then f -1(b) = a. • If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1. If a function is one-to-one, the graph of its inverse f -1(x) is a reflection of the graph of f across the line y = x. Copyright © 2007 Pearson Education, Inc. Slide 5-8 5.1 Finding the Inverse of a Function with a Restricted Domain Example Let f ( x) x 5. Find f 1 ( x). Solution Notice that the domain of f is restricted to [–5,), and its range is [0, ). It is one-to-one and thus has an inverse. y x5 x y5 x2 y 5 y x2 5 The range of f is the domain of f -1, so its inverse is f 1 ( x) x 2 5, x 0. Copyright © 2007 Pearson Education, Inc. Slide 5-9 5.1 Important Facts About Inverses 1. If f is one-to-one, then f -1 exists. 2. The domain of f is the range of f -1, and the range of f is the domain of f -1. 3. If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1, so the graphs of f and f -1 are reflections of each other across the line y = x. Copyright © 2007 Pearson Education, Inc. Slide 5-10 5.1 Application of Inverse Functions Example Use the one-to-one function f(x) = 3x + 1 and the numerical values in the table to code the message BE VERY CAREFUL. A 1 F 6 K 11 P 16 U 21 B 2 G 7 L 12 Q 17 V 22 C 3 H 8 M 13 R 18 W 23 D 4 I 9 N 14 S 19 X 24 E 5 J 10 O 15 T 20 Y 25 Z 26 Solution BE VERY CAREFUL would be encoded as 7 16 67 16 55 76 10 4 55 16 19 64 37 because B corresponds to 2, and f(2) = 3(2) + 1 = 7, and so on. Copyright © 2007 Pearson Education, Inc. Slide 5-11