Function Composition

Definition and Notation

Function composition is a fundamental concept in algebra where two functions are combined to form a new function. The composition of functions f and g, denoted as (f ˆ g)(x), means applying g first and then applying f to the result of g(x).

• Notation:

• Domain: must be in the domain of , and must be in the domain of .

Example: If and , then:

Evaluating Compositions at Specific Values

To evaluate a composition at a specific value, substitute the value into the inner function and then apply the outer function.

• Example:

• First, find , then substitute that result into .

Domain of Composed Functions

The domain of a composed function consists of all in the domain of such that is in the domain of .

• Check the domain of .

• Check that outputs values in the domain of .

Function Operations

Combining Functions

Functions can be combined using addition, subtraction, multiplication, and division, as well as composition.

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

Examples of Function Operations

• Given:

• Domain:

• Given:

• Domain: All real numbers (since any real can be used in a polynomial and addition).

Summary Table: Function Composition and Operations

Additional info:

• When composing functions, always pay attention to the order: in general.

• For radical and rational functions, domain restrictions are especially important.