1. Slide 2.8- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inverse Functions Learn the definition of an inverse function and a relation. Learn to identify one-to-one functions. Learn a procedure for finding an inverse function. Learn to use inverse functions to find the range of a function. Learn to apply inverse functions in the real world. SECTION 2.8 1 2 3 4 5

3. Slide 2.8- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Determining whether an Inverse Relation is a Function Ray’s Music Mart has six employees. The first table lists the first names and the Social Security numbers of the employees, and the second table lists the first names and the ages of the employees. a.Find the inverse of the function defined by the first table, and determine whether the inverse relation is a function. b.Find the inverse of the function defined by the second table, and determine whether the inverse relation is a function.

4. Slide 2.8- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Determining whether an Inverse Relation is a Function Solution Every y–value corresponds to exactly one x– value. Thus the inverse of the function defined in this table is a function. Dwayne590-56-4932 Sophia599-23-1746 Desmonde264-31-4958 Carl432-77-6602 Anna195-37-4165 Sal543-71-8026

5. Slide 2.8- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Determining whether an Inverse Relation is a Function Solution continued There is more than one x– value that corresponds to a y–value. For example, the age of 24 yields the names Dwayne and Anna. Thus the inverse of the function defined in this table is not a function. Dwayne24 Sophia26 Desmonde42 Carl51 Anna24 Sal26

6. Slide 2.8- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

7. Slide 2.8- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A ONE-TO-ONE FUNCTION Each y-value in the range corresponds to only one x-value in the domain.

8. Slide 2.8- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

9. Slide 2.8- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley NOT A FUNCTION The x-value x 2 in the domain corresponds to the two y-values, y 2 and y 3, in the range.

10. Slide 2.8- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HORIZONTAL- LINE TEST A function f is one-to-one if no horizontal line intersects the graph of f in more than one point.

11. Slide 2.8- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 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.

12. Slide 2.8- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 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.

13. Slide 2.8- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 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.

14. Slide 2.8- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF f –1 FOR A ONE-TO-ONE FUNCTION f Let f represent a one-to-one function. The inverse of f is also a function, called the inverse function of f, and is denoted by f –1. If (x, y) is an ordered pair of f, then (y, x) is an ordered pair of f –1, and we write x = f –1 (y). We have y = f (x) if and only if f –1 (y) = x.

15. Slide 2.8- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 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.

16. Slide 2.8- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

17. Slide 2.8- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley UNIQUE INVERSE FUNCTION PROPERTY 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

18. Slide 2.8- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Verifying Inverse Functions Verify that the following pairs of functions are inverses of each other: Solution Form the composition of f and g.

19. Slide 2.8- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Verifying Inverse Functions Solution continued Now find Sincewe conclude that f and g are inverses of each other.

20. Slide 2.8- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

21. Slide 2.8- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 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

22. Slide 2.8- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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).

23. Slide 2.8- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding the Inverse Function Find the inverse of the one-to-one function Solution Step 1 Step 2 Step 3

24. Slide 2.8- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding the Inverse Function Solution continued Step 4 Step 3 (cont.)

25. Slide 2.8- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Finding the Domain and Range Find the domain and the range of the function Solution Domain of f, all real numbers x such that x ≠ 2, in interval notation (–∞, 2) U (2, –∞). Range of f is the domain of f –1. Range of f is (–∞, 1) U (1, –∞).

26. Slide 2.8- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

27. Slide 2.8- 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding an Inverse Function Solution continued Here are the graphs of g and g –1.

28. Slide 2.8- 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

29. Slide 2.8- 29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.