Surjective function

A surjective function is a function $f:A\to B$ such that for every $b\in B$ there exists at least one $a\in A$ with $f(a)=b$.

Surjectivity can be expressed using the image : $f$ is surjective exactly when $f(A)$ equals its codomain $B$. Surjectivity is one of the two conditions (along with injectivity) needed for a bijection .

Examples:

• The function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3$ is surjective, since every real number has a real cube root.
• The parity map $p:\mathbb{Z}\to\{0,1\}$ (even $\mapsto 0$, odd $\mapsto 1$) is surjective.