1. The Composition of Functions
Section 2.3
The Composition of Functions

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) = 2x  5 and g(x) = x2 − 3x + 8, find: and

5. Example continued
Given that f(x) = 2x  5 and g(x) = x2 − 3x + 8, find: and .

6. Example
Given :
Find
Find the domain of

7. Example continued Given : Find Find the domain of
b) The domain of f(x) is {x | x ≥ 0}, or [0, ∞). Any real number can be an input for g(x), so the domain of g(x) is (−∞,∞).
The domain of consists of the values of x in the domain of g, for which g(x) is nonnegative.

8. Example continued
Thus we have
We see that the domain of

9. Example continued
The domain of consists of the values of x in the domain of f, [0, ∞), for which g(x) is defined. Since g can accept any real number as an input, any output from f is acceptable, so the entire domain of f is the domain of
. That is, the domain of is

10. Decomposing a Function as a Composition
In calculus, we often need to recognize how a function can be expressed as the composition of two functions. This can be thought of as “decomposing” the function.

11. Example
If h(x) = (2x  3)5, find f(x) and g(x) such that The function h(x) raises (2x  3) to the fifth power. Two functions that can be used for the composition are: f(x) = x5 and g(x) = 2x  3.