1 1.6: Inverse functions. Find the inverse of the function and algebraically verify they are inverses.

Copyright © Cengage Learning. All rights reserved. Pre-Calculus Honors 1.6: Inverse Functions HW: p.67 (12, all, 22, 28, 34, even) Tomorrow: p.68 (36, all, 50, even, 115) Test : Thursday

3 Copyright © Cengage Learning. All rights reserved. Review for Test : Review all notes, worksheet, assigned homework, and quiz. Supplemental review below: p.86 (all) p.68 (57, 58) p.82 – 85 (27, 45, 47, 49, odd, odd, odd, odd, odd, 147)

4 Which of the functions is the inverse of ? or Verify algebraically.

5 The Graph of an Inverse Function The graphs of a function and its inverse function f –1 are related to each other in the following way. If the point (a, b) lies on the graph of then the point (b, a) must lie on the graph of f –1 and vice versa. This means that the graph of f –1 is a reflection of the graph of f in the line y = x as shown in Figure Figure 1.57

6 Sketch the graph of f -1.

7 Example 5 – Verifying Inverse Functions Graphically Verify that the functions f and g are inverse functions of each other graphically and numerically. Solution: From Figure 1.58, you can conclude that f and g are inverse functions of each other. Figure 1.58

8 The Existence of an Inverse Function To have an inverse function, a function must be one-to-one, which means that no two elements in the domain of f correspond to the same element in the range of f. (Note: In order for a relation to be a function every element in the domain corresponds to one unique element in the range. Every input corresponds to one output.)

9 One-to-One From figure 1.61, it is easy to tell whether a function of x is one-to-one. Simply check to see that every horizontal line intersects the graph of the function at most once. This is called the Horizontal Line Test. f (x) = x 2 is not one-to-one. Figure 1.61

10 Determine algebraically whether the function is 1-to-1. If f(a) = f(b) implies a = b, then the function is one- to-one and it does have an inverse function.

11 Determine algebraically whether the function has an inverse. 1.) 2.) 3.)4.)