1. The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function f with the function g is defined by (f ○ g)(x) = f(g(x)). Two step process to find y = f(g(x)): 1. Find h = g(x). 2. Find y = f(h) = f(g(x)) Composite Functions

2. A function, f, has an inverse function, g, if and only if f ◦ g(x) = x and g ◦ f(x)) = x, for every x in domain of g and in the domain of f. Definition of an Inverse Function

3. Example 1 Name two points on the inverse of the function t, when t(x) = x 3 + 5x + 7 Plug in relatively easy numbers! (0,7), (1,13) Now flip them!! (7,0) and (13,1)

4. Is the inverse a function? 1.If the relation passes the vertical line test, it is a function. 2.If the relation passes the horizontal line test, its inverse is a function.

5. 1. Given the function y = f(x). 2. Interchange x and y. 3. Solve the result of Step 2 for y = g(x). 4. If y = g(x) is a function, then g(x) = f -1 (x). Finding the Inverse of a Function

6. Example 2 y = 18 – x 2 x = 18 – y 2 x – 18 = -y 2 -x + 18 = y 2

7. Find the inverse and then verify that the two functions are inverses of each other.

8. f ◦ g =

9. g ◦ f =

10. Homework Pages 212 - 213 2, 5 -13

11. Thanks to: Dr. Claude S. Moore Danville Community College