Chapter 5: Inverse, 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

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.

5.1 One-to-One Functions Only functions that are one-to-one have inverses. 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).

5.1 One-to-One Functions Example Decide whether each 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

5.1 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

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 Example are inverse functions of each other.

5.1 Finding an Equation for the Inverse Function Notation for the inverse function f -1 is read “f-inverse”. By the definition of inverse function, the domain of f equals the range of f -1, and the range of f equals the domain of f -1.

5.1 Finding an Equation for the Inverse Function Finding the Equation of the Inverse of y = f(x) For a one-to-one function f defined by an equation y = f(x), find the defining equation of the inverse as follows. (Any restrictions on x and y should be considered.) 1. Interchange x and y. 2. Solve for y. 3. Replace y with f -1(x).

5.1 Example of Finding f -1(x) Example Find the inverse, if it exists, of Solution Write f (x) = y. Interchange x and y. Solve for y. Replace y with f -1(x).

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.

5.1 Finding the Inverse of a Function with a Restricted Domain Example Let 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. The range of f is the domain of f -1, so its inverse is

5.1 Important Facts About Inverses If f is one-to-one, then f -1 exists. The domain of f is the range of f -1, and the range of f is the domain of f -1. 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.

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.