1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

2. OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Inverse Functions SECTION 2.9 1 2 3 4 Define an inverse function. Find the inverse function. Use inverse functions to find the range of a function. Apply inverse functions in the real world.

3. 3 © 2010 Pearson Education, Inc. All rights reserved DEFINITION OF A ONE-TO-ONE FUNCTION A function is called a one-to-one function if each y-value in its range corresponds to only one x-value in its domain.

4. 4 © 2010 Pearson Education, Inc. All rights reserved A ONE-TO-ONE FUNCTION Each y-value in the range corresponds to only one x-value in the domain.

5. 5 © 2010 Pearson Education, Inc. All rights reserved NOT A ONE-TO-ONE FUNCTION The y-value y 2 in the range corresponds to two x-values, x 2 and x 3, in the domain.

6. 6 © 2010 Pearson Education, Inc. All rights reserved NOT A FUNCTION The x-value x 2 in the domain corresponds to two y-values, y 2 and y 3, in the range.

7. 7 © 2010 Pearson Education, Inc. All rights reserved HORIZONTAL- LINE TEST A function f is one-to-one if no horizontal line intersects the graph of f in more than one point.

8. 8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Horizontal-Line Test Use the horizontal-line test to determine which of the following functions are one-to-one. Solution No horizontal line intersects the graph of f in more than one point, therefore the function f is one-to-one.

9. 9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Horizontal-Line Test Solution continued There are many horizontal lines that intersect the graph of f in more than one point, therefore the function f is not one-to-one.

10. 10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Horizontal-Line Test Solution continued No horizontal line intersects the graph of f in more than one point, therefore the function f is one-to-one.

11. 11 © 2010 Pearson Education, Inc. All rights reserved DEFINITION OF AN INVERSE FUNCTION f Let f represent a one-to-one function. Then if y is in the range of f there is only one value of in the domain of f such that f(x) = y. We define the inverse of f, called the inverse function of f, denoted f –1, by f –1 (y) = x if and only if y = f(x).

12. 12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Relating the Values of a Function and Its Inverse Assume that f is a one-to-one function. a.If f (3) = 5, find f –1 (5). b.If f –1 (–1) = 7, find f (7). a.Let x = 3 and y = 5. Now 5 = f (3) if and only if f –1 (5) = 3. Thus, f –1 (5) = 3. b.Let y = –1 and x = 7. Now, f –1 (–1) = 7 if and only if f (7) = –1. Thus, f (7) = –1. Solution We know that y = f (x) if and only if f –1 (y) = x.

13. 13 © 2010 Pearson Education, Inc. All rights reserved INVERSE FUNCTION PROPERTY Let f denote a one-to-one function. Then for every x in the domain of f –1. 1. for every x in the domain of f. 2.

14. 14 © 2010 Pearson Education, Inc. All rights reserved Let f denote a one-to-one function. Then if g is any function such that g = f –1. That is, g is the inverse function of f. for every x in the domain of g and for every x in the domain of f, then INVERSE FUNCTION PROPERTY

15. 15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Verifying Inverse Functions Verify that the following pairs of functions are inverses of each other: Solution Form the composition of f and g.

16. 16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Verifying Inverse Functions Solution continued Now form the composition of g and f. Since we conclude that f and g are inverses of each other.

17. 17 © 2010 Pearson Education, Inc. All rights reserved SYMMETRY PROPERTY OF THE GRAPHS OF f AND f –1 The graph of the function f and the graph of f –1 are symmetric with respect to the line y = x.

18. 18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Graph of f –1 from the Graph of f The graph of the function f is shown. Sketch the graph of the f –1. Solution

19. 19 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR FINDING f –1 Step 1Replace f (x) by y in the equation for f (x). Step 2Interchange x and y. Step 3Solve the equation in Step 2 for y. Step 4Replace y with f –1 (x).

20. 20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Finding the Inverse Function Find the inverse of the one-to-one function Solution Step 1 Step 2 Step 3

21. 21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Finding the Inverse Function Solution continued Step 4 Step 3 (cont.)

22. 22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Domain and Range Find the domain and the range of the function Solution Domain of f is the set of all real numbers x such that x ≠ 2. In interval notation that is (–∞, 2) U (2, –∞). Range of f is the domain of f –1. Range of f is (–∞, 1) U (1, –∞).

23. 23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding an Inverse Function Find the inverse of G(x) = x 2 – 1, x ≥ 0. Solution Step 1 y = x 2 – 1, x ≥ 0 Step 2 x = y 2 – 1, y ≥ 0 Step 4 Step 3 Since y ≥ 0, reject

24. 24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding an Inverse Function Solution continued Here are the graphs of G and G –1.

25. 25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Water Pressure on Underwater Devices The formula for finding the water pressure p (in pounds per square inch), at a depth d (in feet) pressure gauge on a diving bell breaks and shows a reading of 1800 psi. Determine how far below the surface the bell was when the gauge failed. below the surface isSuppose the Solution The depth is given by the inverse of

26. 26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Water Pressure on Underwater Devices Solution continued Solve the equation for d. Let p = 1800. The device was 3960 feet below the surface when the gauge failed.