EXAMPLE 1 Find an inverse relation Find an

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

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

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

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

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

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

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

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

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