5.3- Inverse Functions If for all values of x in the domains of f and g, then f and g are inverse functions.

Prove f and g are inverse functions. f(x) = -2x + 6 g(x) = - ½ x + 3

Prove f and g are inverse functions.

For each f(x), find the inverse function, f-1(x) b. c.

d. e.

Analyzing Inverses f(x) = 3x - 6 a. Find 5 points in a data table. b. Graph f. c. Find 5 points in a data table of the inverse of f(x). d. Find f-1 (Be careful that you find the correct inverse.) e. Graph the inverse of f(x) or f-1 (x).

f(x) = x3 a. Find 5 points in a data table. b. Graph f. c. Find f-1. d. Find 5 points in a data table. e. Graph f-1.

f(x) = x2 a. Find 5 points in a data table. b. Graph f. c. Use the data table above to find five points for the inverse of f. d. Plot the points that represent the inverse, then graph the inverse of f(x).

e. Describe the domain and range of f(x) Domain __________ Range __________ f. Describe the domain and range of f-1(x) g. What can you say about the domain and range of a function and its inverse? h. Is the inverse of f(x) a function? How do you know? (Look at the graph.)

If f and g are inverse functions, a. the graph of g is a reflection image of f over the line y = x. b. the domain of f is the range of g and the range of f is the domain of g. c. for all x in the domain of f, g(f(x))=x and for all x in the domain of g, f(g(x))=x.

For f(x) to have an inverse function g(x), f(x) and g(x) must be: ONE-to-ONE functions A one-to-one function has exactly one output (y) for each input (x) AND exactly one input (x) for each output (y). One-to-one functions will pass the vertical line test and the horizontal line test.