Types of Functions

The definitions, building on each other

A function $f$ is one-one (injective) if different inputs always give different outputs, and onto(surjective) if every element of the codomain actually gets hit by something. A function that's both is called bijective. Bijective is exactly the condition a function needs to be invertible: to have a genuine inverse function that undoes it.

Worked example: checking one-one and onto

Let $f: \mathbb{N} \to \mathbb{N}$ be defined by $f(n) = \dfrac{n+1}{2}$ if $n$ is odd, and $f(n) = \dfrac{n}{2}$ if $n$ is even. Is $f$ one-one, onto, both, or neither?

Solution: Check a couple of small inputs: $f(1) = \tfrac{1+1}{2} = 1$ and $f(2) = \tfrac{2}{2} = 1$. Two different inputs, $1$ and $2$, give the same output, so $f$ is not one-one. But for any target $m \in \mathbb{N}$, the even input $n=2m$ gives $f(2m) = m$, so every output is reachable, and $f$ is onto.

Worked example: finding an inverse

$f: \mathbb{N} \to \mathbb{R}$ is defined by $f(x) = 4x^2+12x+15$. Show $f: \mathbb{N} \to S$ (where $S$ is the range of $f$) is invertible, and find the inverse.

Solution: Complete the square first:

Since $x \in \mathbb{N}$ means $x+\tfrac{3}{2} > 0$, this squaring step is reversible: each $y$ comes from exactly one $x$, so $f$ is invertible on its range. Solving for $x$:

So $f^{-1}(y) = \dfrac{\sqrt{y-6}-3}{2}$.

Worked example: a reflection is an inverse

$f(x) = (x+1)^2$ for $x \ge -1$. If $g(x)$ is the reflection of $f(x)$ in the line $y=x$, find $g(x)$.

Solution: Reflecting a graph in $y=x$ is exactly what taking the inverse function does, so $g = f^{-1}$. Setting $y=(x+1)^2$ and solving for $x$ (taking the non-negative root since $x+1 \ge 0$):