Composition of Functions

Composition of Functions

## Definition / Concept

If:

$$f:A\to B$$

and:

$$g:B\to C$$

then $g\circ f$ maps $A$ to $C$.

## Formula

$$(g\circ f)(x)=g(f(x))$$

$$g\circ f:A\to C$$

## Conditions / Special Cases

For the composition to be defined:

$$\text{Range}(f)\subseteq\text{Domain}(g)$$

In general:

$$g\circ f\ne f\circ g$$

Composition is associative:

$$h\circ(g\circ f)=(h\circ g)\circ f$$

## Usage

- Used in function equations and inverse-function problems.

- Useful for simplifying nested functions.