Types of Functions

There are three ($3$) types of functions that we study at this level:

• Injective functions
• Surjective functions
• Bijective functions

Injective (one-to-one) functions

For a function to be injective, every $x$ value must have exactly one ($1$) $y$ value. In other words, every output has a unique input. Injective functions can be referred to as injections.

The algebraic method of determining injectivity

Given a function $f(x)$, we let $$ \begin{equation}\begin{aligned} f(a)=f(b)\\ \end{aligned}\end{equation} $$ If we can prove exactly that $$ \begin{equation}\begin{aligned} a=b\\ \end{aligned}\end{equation} $$ Then the function $f(x)$ is injective. Note that a variation such as $a=\pm b$ will disqualify the function from being injective. We must be able to show exactly that $a=b$.

The horizontal line test

In this test we simply draw a horizontal line to intersect the graph of our function. If it cuts/intersects more than once, the function is not injective. This is because multiple intersections implies that multiple $x$ values can produce the same $y$ value.

Surjective (onto) functions

For a function to be surjective, there must be at least one $x$ value for every $y$ value. In other words, every output has a corresponding input. We write: $$ \begin{equation}\begin{aligned} f:X\rightarrow Y\text{ is surjective if }\\ \forall y\in Y, \exists x\in X\text{ s.t. } f(x)=y\\ \end{aligned}\end{equation} $$

This means that if we can find any $y$ value in the co-domain $Y$ that does not produce a defined $x$ value, the function cannot be surjective onto the co-domain $Y$. A function can thus be surjective given one co-domain but not surjective given another co-domain. Surjective functions can be referred to as surjections.

$\forall$ is known as the universal quantifier. It is read as “for every”. Thus the notation $\forall y\in Y$ is read as “for every value $y$ in $Y$”. $\exists$ is known as the existential quantifier. It is read as “there exists at least one”. Thus the notation $\exists x\in X$ is read as “there exists at least one $x$ in the set $X$”. s.t. means “such that”.

Bijective functions

These are functions which are both injective (one-to-one) and surjective (onto).

Only bijective functions have inverses. A function has an inverse if and only if it is bijective.