1

1.6: Inverse functions.

12)( 3 xxf

Find the inverse of the function and algebraically verify they are inverses.

Copyright © Cengage Learning. All rights reserved.

Pre-Calculus Honors1.6: Inverse Functions

HW: p.67 (12, 15-18 all, 22, 28, 34, 94-100 even)Tomorrow: p.68 (36, 39-44 all, 50, 60-70 even, 115)

Test 1.1-1.7: Thursday

3Copyright © Cengage Learning. All rights reserved.

Review for Test 1.1-1.7: 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, 55-69 odd,

79-85 odd, 93-103 odd, 107-115 odd, 127-141 odd, 147)

4

25)(

x

xfWhich of the functions is the inverse of ?

or

Verify algebraically.52)(

xxg 25)(

xxh

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 off –1 is a reflection of the graph of f in the line y = x as shown in Figure 1.57.

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 eachother.

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) = x2 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.)

4)( xxf 543)(

xxf

32)( xxf 2;2)( xxxf