Types of Functions — One-One, Onto, Into and Bijective

Once we know a rule $f : A \to B$ is a function, we classify how it matches elements of $A$ to elements of $B$. Four standard names cover this classification.

One-one (injective) function. A function $f : A \to B$ is one-one if different elements of $A$ always have different images — no two distinct inputs are ever allowed to share the same output. Formally,

which is equivalent to its contrapositive $x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$. The standard algebraic test is to assume $f(x_1) = f(x_2)$ and show, by valid algebra, that this forces $x_1 = x_2$.

Onto (surjective) function. A function $f : A \to B$ is onto if every element of the codomain $B$ is the image of at least one element of $A$ — i.e. the range equals the whole codomain. Formally, for every $b \in B$ there exists at least one $a \in A$ with $f(a) = b$. The standard test is to take an arbitrary $b$ in the codomain, solve $f(x) = b$ for $x$, and confirm a valid $x$ in the domain always exists.

Into function. If $f : A \to B$ is not onto — the range is a proper subset of the codomain, leaving at least one element of $B$ with no pre-image — then $f$ is called an into function.

Bijective function (one-one correspondence). A function that is both one-one and onto is bijective. It pairs every element of $A$ with a distinct element of $B$, using up the whole of $B$, with nothing left over on either side — exactly the condition a function needs before it can have a genuine inverse function (Section 7).

A function that is not one-one is called many-one; combining the two dials gives four possibilities: