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 = y= y

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

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

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 – Add 5 to each side. R + 5 = 3 8 L R += L Multiply each side by 8 3. SOLUTION

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

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

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 = y= y for Examples 1, 2, and 3

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

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 = R = (13) = 48 ANSWER The band provides 13 pounds of resistance when it is stretched to 48 inches. for Examples 1, 2, and 3