Existence of inverse functions

Introduction to inverse functions

An inverse function reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it.

We denote the inverse of $f$ by $f^{-1}.$

For example, if $f(x)=2x+1,$ and $x=1$ we can make use of the function to find $y=f(1)=3.$ The inverse function aims to take the value $y=3,$ and get back the original value of $x=1.$ We can check that the function $f^{-1}(x)=\frac{x-1}{2}$ achieves this aim (we will learn how to find this in a later section).

Inverse functions may not exist

Consider the function $g(x)=x^2.$ It is a function: every input $x$ leads to exactly one output $g(x).$ However, the reverse is not true: an output of $y=4$ could have come from either $x=-2$ or $x=2.$ In such a case we say that the inverse function does not exist.

One-to-one functions

The earlier function $f$ sends each input to a different output while $g$ will sometimes sends different inputs to the same output. This is what allows $f$ to have an inverse while $g$ does not.

We call this behavior of sending each input to a different output a one-to-one function.

We can check if a function is one-to-one (and thus its inverse exist) by employing the horizontal line test.

The following two examples illustrate how to phrase the horizontal line test for the cases where $f$ is one-to-one and has an inverse, and where $f$ is not one-to-one and does not have an inverse.