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Copyright © 2007 Pearson Education, Inc. Slide 5-2 Chapter 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

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Copyright © 2007 Pearson Education, Inc. Slide 5-3 5.1 Inverse Functions Example Also, f [g(12)] = 12. For these functions, it can be shown that for any value of x. These functions are inverse functions of each other.

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Copyright © 2007 Pearson Education, Inc. Slide 5-4 Only functions that are one-to-one have inverses. 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, a b implies f (a) f (b).

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Copyright © 2007 Pearson Education, Inc. Slide 5-5 5.1 One-to-One Functions Example Decide whether the function is one-to-one. (a) (b) Solution (a) For this function, two different x-values produce two different y-values. (b)If we choose a = 3 and b = –3, then 3 –3, but

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Copyright © 2007 Pearson Education, Inc. Slide 5-6 5.1 The Horizontal Line Test Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions. (a) (b) If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Not one-to-one One-to-one

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Copyright © 2007 Pearson Education, Inc. Slide 5-7 5.1 Inverse Functions Example are inverse functions of each other. Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

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Copyright © 2007 Pearson Education, Inc. Slide 5-8 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.

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Copyright © 2007 Pearson Education, Inc. Slide 5-9 5.1 Example of Finding f -1 (x) ExampleFind the inverse, if it exists, of Solution Write f (x) = y. Interchange x and y. Solve for y. Replace y with f -1 (x).

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Copyright © 2007 Pearson Education, Inc. Slide 5-10 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.

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Copyright © 2007 Pearson Education, Inc. Slide 5-11 5.1 Finding the Inverse of a Function with a Restricted Domain ExampleLet SolutionNotice that the domain of f is restricted to [–5, ), and its range is [0, ). It is one-to-one and thus has an inverse. The range of f is the domain of f -1, so its inverse is

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Copyright © 2007 Pearson Education, Inc. Slide 5-12 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.

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Copyright © 2007 Pearson Education, Inc. Slide 5-13 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. A1F6K 11P 16U21 B 2G 7L 12Q 17V22 C 3H 8M 13R 18W23 D4I 9N 14S 19X24 E 5J 10O 15 T 20Y 25 Z 26 SolutionBE 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.

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