Composition of Functions

Definition and Properties

Similarly to relations, we can compose two or more functions to create a new function. This operation is called the composition of functions.

Let g : A → B and f : B → C be two functions such that the range of g equals the domain of f. The composition of the functions f and g, denoted by f∘g, is another function defined as

Since functions are a special case of relations, they inherit all properties of composition of relations and have some additional properties. We list here some of them:

1. The composition of functions is associative. If $$ $$ and $$ then $$
2. The composition of functions is not commutative. If $$ and $$ then, as a rule, $$
3. Let $$ and $$ be injective functions. Then the composition of the functions $$ is also injective.
4. Let $$ and $$ be surjective functions. Then the composition of the functions $$ is also surjective.
5. It follows from the last two properties that if two functions $$ and $$ are bijective, then their composition $$ is also bijective.

Examples

Example 1. Composition of Functions Defined on Finite Sets

Consider the sets $$ $$ and $$ The functions $$ and $$ are defined as

It is convenient to illustrate the mapping between the sets in an arrow diagram:

Given the mapping, we see that

Hence, the composition of functions $$ is given by

This is represented in the following diagram:

Example 2. Composition of Functions Defined on Infinite Sets

Let $$ and $$ be two functions defined as

Determine the composite functions $$ $$ $$ $$

The first composite function $$ is formed when the inner function $$ is substituted for $$ in the outer function $$ This yields:

Similarly we find the other composite functions:

Compositions Involving Inverse Functions

Let $$ be a bijective function from domain $$ to codomain $$ Then it has an inverse function $$ that maps $$ back to $$ Then

where $$ is the identity function in the domain $$ and $$ is any element of $$

Similarly,

where $$ is the identity function in the codomain $$ and $$ is any element of $$