1. EXAMPLE 1 Find an inverse relation Find an equation for the inverse of the relation y = 3x – 5. Write original relation. y = 3x – 5 Switch x and y. x = 3y – 5 Add 5 to each side. x + 5 = 3y Solve for y. This is the inverse relation. 1 3 x +x + 5 3 = y= y

2. EXAMPLE 2 Verify that functions are inverses Verify that f(x) = 3x – 5 and f –1 (x) = 1 3 x + 5 3 are inverse functions. STEP 1 Show: that f(f –1 (x)) = x. f (f –1 (x)) = f 3 1 x + 5 3 = x + 5 – 5 = x SOLUTION 3 1 x + 5 3 = 3 – 5 STEP 2 Show: that f –1 (f(x)) = x. = 1 3 5 3 (3x – 5) + = x – 5 3 5 3 + = x f –1 (f(x)) = f –1 (3x – 5)

3. EXAMPLE 3 Solve a multi-step problem 3 8 Elastic bands can be used in exercising to provide a range of resistance. A band’s resistance R (in pounds) can be modeled by R = L – 5 where L is the total length of the stretched band (in inches). Fitness

4. EXAMPLE 3 Solve a multi-step problem Use the inverse function to find the length at which the band provides 19 pounds of resistance. Find the inverse of the model. STEP 1 Find: the inverse function. Write original model. R = L – 5 3 8 Add 5 to each side. R + 5 = 3 8 L 8 3 40 3 R += L Multiply each side by 8 3. SOLUTION

5. EXAMPLE 3 Solve a multi-step problem STEP 2 Evaluate: the inverse function when R = 19. 40 3 L = 8 3 + R + 8 3 = (19) + 40 3 3 152 3 = + 192 3 = = 64 ANSWER The band provides 19 pounds of resistance when it is stretched to 64 inches.

6. for Examples 1, 2, and 3 GUIDED PRACTICE Find the inverse of the given function. Then verify that your result and the original function are inverses. 1. f(x) = x + 4 Write original relation. y = x + 4 Switch x and y. x = y + 4 Subtract 4 from each side. x – 4 = y

7. GUIDED PRACTICE 2. f(x) = 2x – 1 Write original relation. y = 2x – 1 Switch x and y. x = 2y – 1 Add 1 to each side. x + 1 = 2y Divide both sides by 2. x + 1 2 = y= y for Examples 1, 2, and 3

8. GUIDED PRACTICE 3. f(x) = –3x – 1 Write original relation. y = –3x + 1 Switch x and y. x = –3y +1 Subtract 1 to each side. x – 1 = –3y Solve for y. This is the inverse relation. for Examples 1, 2, and 3 x  1 33 = y= y

9. GUIDED PRACTICE 4. Fitness: Use the inverse function in Example 3 to find the length at which the band provides 13 pounds of resistance. Evaluate the inverse function when R = 3 SOLUTION 40 3 L = 8 3 + R + 8 3 = (13) + 40 3 = 48 ANSWER The band provides 13 pounds of resistance when it is stretched to 48 inches. for Examples 1, 2, and 3