# Inverse Functions

## Introduction

One-to-one functions, or invertible functions, are functions that have inverses. One-to-one functions have a nice test to tell from the graph if a function is one-to-one. The definition of a one-to-one function is given through the equation of the function. The conceptual idea for a function being one-to-one is that for every number in the range there is one number in the domain that the function maps to the point in the range.

## Definition

A function is one-to-one if two distinct values in the domain get mapped to the same value in the range. More concretely, if f is one-to-one and are two numbers in the domain of f such that then we must have that

Examples:

Non-examples:

## Horizontal Line Test

It was mentioned earlier that there is a way to tell if a function is one-to-one from its graph. This method is called the horizontal line test. It is the same as the vertical line test, except we use a horizontal line. So a function is one-to-one if every horizontal line crosses the graph at most once.

## Inverse Functions

If a function is one-to-one, then to each y in the range of f there is a unique x in the domain that maps on top of it. So we can define a function from the range of f back to the domain. This new function is called the inverse function, and is denoted .

Properties: Let f be a one-to-one function, and be its inverse. a) Domain of f = Range of b) Range of f = Domain of c) where x is in the domain of f^{-1} d) where x is in the domain of

Since we have both a graphical and algebraic method to determine if a function is one-to-one, we also have geometric and algebraic methods to find the inverse.

The graphical method is to reflect the graph of f across the line y = x. Although easy to explain, the process is harder to visualize. The algebraic method is the opposite, harder to explain but easier to execute.

The algebraic method can be explained in 3 steps:

Step 1) In the equation y = f(x), interchange all instances of x with the variable y, and vice versa.

Step 2) If possible, solve for y.

Step 3) Replace y with and check that

Example: Find the inverse of

We start by swapping the x and y to get

Now we can solve for y and swap y for . Doing so we will find that