1. The Composition of Functions
Section 2.3
The Composition of Functions
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

2. Objectives
Find the composition of two functions and the domain of the composition.
Decompose a function as a composition of two functions.

3. Composition of Functions
Definition:

4. Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b) a)

5. Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find:
a) b) b)

6. Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b) a)

7. Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find:
a) b) b)

8. Example Given , find the domain of
f (x) is not defined for negative radicands. Since the inputs of are the outputs of g, the domain of consists of all the values in the domain of g for which g(x) is nonnegative.
The domain is

9. Decomposing a Function as a Composition
In calculus, one needs to recognize how a function can be expressed as the composition of two functions. This can be thought of as “decomposing” the function.

10. Example
If h(x) = (3x  1)4, find f(x) and g(x) such that The function h(x) raises (3x  1) to the fourth power. Two functions that can be used for the composition are: f(x) = x4 and g(x) = 3x  1.