Inverse functions:

1. Inverse functions: • if f is one-to-one function with domain X and range Y and g is function with domain Y and range X then g is the inverse function of f if and only if • (f o g)(x)= x for all x in domain of g • (g o f)(x)= x for all x in domain of f

2. Verify functions are inverses g(x) = 1/2 x – 4 f(x) = 2x + 8 (f o g)(x) = 2(1/2 x - 4) + 8 = x - 8 + 8 = x (g o f)(x) = 1/2(2x + 8) - 4 = x + 4 - 4 = x

3. Inverse notation • f -1is the inverse function notation (not the reciprocal) • function f has inverse if and only if f is aone-to-onefunction

4. Find the inversef -1 1. Substitute y for f(x) 2. Interchange x and y 3. Solve if possible for y in terms of x 4. Substitute f -1 (x) for y 5. Verify domain of f is range of f -1

5. find inverse of f(x) = 2x - 6 f(x) = 2x - 6 y = 2x - 6 x = 2y - 6 x + 6 = 2y y = 1/2 x + 3 f -1 (x) = 1/2 x + 3 f has D & R of all reals f -1 also has D & R of all reals

6. find inverse of function Inverse of ordered pairs, f(x) = {(1,2), (3,4), (5,6)} 1–1 function? f -1(x) = {(2,1), (4,3), (6,5)} • inverse is a function g(x) = {(1,2), (5,4), (3,2)}, 1–1 function? g -1(x) = {(2,1), (4,5), (2,3)} • inverse is not a function