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

Once we know a rule $f: A \to B$ is a function, we can further classify how it matches elements of $A$ to elements of $B$. Four names cover the classification used throughout the CHSE Odisha Business Mathematics & Statistics syllabus.

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

(equivalently, its contrapositive: $x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$). Algebraically, the standard way to check one-oneness 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 $a \in A$ such that $f(a)=b$. The standard way to check onto-ness is to take an arbitrary $b$ in the codomain, solve $f(x)=b$ for $x$, and confirm that a valid $x$ in the domain always exists.

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

Bijective function (one-one correspondence). A function that is both one-one and onto is called bijective. A bijective function pairs up every element of $A$ with a distinct element of $B$, using up the whole of $B$ in the process, with nothing left over on either side — this is exactly the condition needed for a function to have a genuine inverse function (Section 6).