A function $f:A\to B$ is a relation that associates every element of the domain $A$ with exactly one element of the codomain $B$.

If $f$ maps $a$ to $b$, we write

$$f(a)=b.$$

Domain, codomain and image

The domain is the set of allowed inputs. The codomain is the set in which outputs are declared to lie.

The image of the function is the subset of the codomain actually reached:

$$f(A)=\lbrace f(a):a\in A\rbrace.$$

Injective functions

A function is injective if distinct inputs have distinct outputs:

$$f(a_1)=f(a_2)\Longrightarrow a_1=a_2.$$

Surjective functions

A function is surjective if every element of the codomain is reached by at least one input.

Bijections

A function that is both injective and surjective is a bijection. A bijection pairs the two sets element-for-element and has an inverse function.

This set-theoretic viewpoint extends the idea of a numerical function: functions can map arbitrary kinds of mathematical objects, not only real numbers.