Onto Function

Onto Function

## Definition / Concept

A function is onto or surjective if every element of the codomain is the image of at least one element of the domain.

## Formula

$$\forall y\in B,\ \exists x\in A\text{ such that }f(x)=y$$

Equivalently:

$$\text{Range}(f)=\text{Codomain}(f)$$

## Conditions / Special Cases

For finite sets:

$$n(A)<n(B)\Rightarrow f:A\to B\text{ cannot be onto}$$

If:

$$n(A)=n(B)$$

then:

$$f\text{ is one-one}\Leftrightarrow f\text{ is onto}$$

## Usage

- Used to test whether the entire codomain is covered.

- Important when determining whether an inverse function exists.