Types of Functions

Exploring different flavours of functions

Jul 24, 2026

Joash Gobin

2-minute read

Made by ChickenFryBytes Studios

Table of Contents

Table of Contents

There are some types of functions that concern us at this level:

Injective (one-to-one) functions
Surjective (onto) functions
Bijective functions

Injective functions

These are functions in which every input produces exactly one unique output. This means that no two $x$ values cannot map onto the same $y$ value.

If we can draw a horizontal line passing through any point of the graph of a function and the line intersects with the graph more than once then the function is disqualified from being injective. Multiple intersections implies that many $x$ values are producing the same $y$ value.

Algebraic method

By letting $$ \begin{equation}\begin{aligned} f(a)=f(b)\\ \end{aligned}\end{equation} $$ if it can be proven strictly that $$ \begin{equation}\begin{aligned} a=b\\ \end{aligned}\end{equation} $$ then the function is injective. Otherwise it must be many-to-one.

Example 1

Prove whether the function $f(x)=3x+2$ is injective. $$ \begin{equation}\begin{aligned} f(a)&=f(b)\\ 3a+2&=3b+2\\ 3a&=3b\\ \therefore a&=b\\ \end{aligned}\end{equation} $$ Because we showed that $a=b$, we conclude that $f(x)=3x+2$ is injective.

Example 2

Prove whether the function $f(x)=x^2-1$ is injective. $$ \begin{equation}\begin{aligned} f(a)&=f(b)\\ a^2-1&=b^2-1\\ a^2&=b^2\\ a&=\pm\sqrt{b^2}\\ a&=\pm b\\ \end{aligned}\end{equation} $$ The result is not strictly $a=b$, hence $f(x)=x^2-1$ is not injective.