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

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 gand in the domain of f.

Definition of an Inverse Function

Example 1

• Name two points on the inverse of the function t, when t(x) = x3 + 5x + 7

Plug in relatively easy numbers! (0,7), (1,13)

Now flip them!! (7,0) and (13,1)

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.

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

Example 2

• y = 18 – x2

• x = 18 – y2

• x – 18 = -y2

• -x + 18 = y2

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

• f ◦ g =

• g ◦ f =

Homework

Pages 212 - 213

2, 5 -13

• Thanks to:

Dr. Claude S. MooreDanville Community

College