The Existence of Inverse Functions

A function has an inverse if and only if it is one-to-one: that is, if for each y-value there is only one corresponding x-value. To test whether or not a function is one-to-one, we can draw multiple horizontal lines through the graph of the function. If any of these horizontal lines intersects with the graph of the function more than once, then the function is NOT one-to-one.

For example, take the graph of the function ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,f(x)=x^{2}}$.

We can draw a horizontal line anywhere on the graph (except for the turning point at ${\displaystyle x=0}$) which will intersect with the graph twice. Therefore f is NOT a one-to-one function, and will NOT have an inverse function.

However, consider the function ${\displaystyle g:\mathbb {R} \to \mathbb {R} ,g(x)=x^{3}}$.

This function IS one-to-one, and will therefore have an inverse function, which we label ${\displaystyle g^{-1}\,}$.

The rule of an inverse function

The rule of the inverse of a function f can be found by letting ${\displaystyle y=f(x)}$, swapping the y and x variables, and re-arranging to make y the subject.

For example, consider g. To find the rule of ${\displaystyle g^{-1}(x)}$, let ${\displaystyle g(x)=y}$ Now ${\displaystyle y=x^{3}\,}$. Swap the x and y variables to get ${\displaystyle x=y^{3}\,}$. Now take the cube root of each side to get ${\displaystyle y={\sqrt[{3}]{x}}\,}$. Now let ${\displaystyle y=g^{-1}(x)}$ and we have:-

${\displaystyle g^{-1}(x)={\sqrt[{3}]{x}}}$

The domain and range of inverse functions

Because, in finding inverse functions, we are swapping the x and y values in each ordered pair, it's logical that the domain of a function is the range of the inverse; and the range of a function is the domain of the inverse.

Thus: ${\displaystyle {\mbox{dom}}f={\mbox{ran}}f^{-1}{\mbox{ and ran}}f={\mbox{dom}}f^{-1}\,}$