Surjective function

Definition

Let $f \colon A \to B$ be a function. The function $f$ is surjective when, for each $b \in B$, there is an $a \in A$ with $f(a) = b$.

Remarks

Other names are onto function and surjection. Equivalently, the range of $f$ is the whole codomain $B$. The function $x \mapsto x^3$ on $\mathbb{R}$ is surjective, and $x \mapsto x^2$ on $\mathbb{R}$ is not, because $-1$ is not a value.

Explanations

L1 human

A surjective function hits every element of the codomain at least once. To prove surjectivity, take an arbitrary $b \in B$ and find an input $a$ with $f(a) = b$.