﻿ Properties of Inverse Functions

# PROPERTIES OF INVERSE FUNCTIONS

• PRACTICE (online exercises and printable worksheets)
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

This lesson will be more accessible if you fully understand the concepts in these prior lessons:
using a function box ‘backwards’
one-to-one functions
undoing a one-to-one function; inverse functions

Every one-to-one function $\,f\,$ has an inverse;
this inverse is denoted by $\,f^{-1}\,$ and read aloud as ‘$\,f\,$ inverse’.

A function and its inverse ‘undo’ each other:   one function does something, the other undoes it.
The purpose of this lesson is to make this idea precise.

## Review: Domain and Range of a Function

The set of allowable inputs for a function $\,f\,$ is called its domain, and is denoted by $\,\text{dom}(f)\,$.

The set of all possible outputs from a function $\,f\,$ is called its range, and is denoted by $\,\text{ran}(f)\,$.
That is, let $\,f\,$ act on every possible input—every element of its domain.
The set of resulting outputs is the range of $\,f\,$: $$\cssId{s19}{\text{ran}(f)} \cssId{s20}{= \{ f(x)\ |\ x\in\text{dom}(f) \}}$$

## The Functions $\,f\,$ and $\,f^{-1}\,$ ‘Undo’ Each Other

 start with an allowable input for $\,f\,$; that is, let $\,x\in\text{dom}(f)\,$ let $\,f\,$ act on $\,x\,$, giving $\,f(x)\,$ let $\,f^{-1}\,$ act on $\,f(x)\,$, giving $\,f^{-1}(f(x))\,$ This returns us to where we started, so: $\,f^{-1}(f(x)) = x\,$ for all $\,x\,$ in the domain of $\,f\,$ In other words, the composite function $\,f^{-1}\circ f\,$ is the identity function on $\,\text{dom}(f)\,$. start with an output from $\,f\,$; that is, let $\,y\in\text{ran}(f)\,$ let $\,f^{-1}\,$ act on $\,y\,$, giving $\,f^{-1}(y)\,$ let $\,f\,$ act on $\,f^{-1}(y)\,$, giving $\,f\bigl(f^{-1}(y)\bigr)\,$ This returns us to where we started, so: $\,f\bigl(f^{-1}(y)\bigr) = y\,$ for all $\,y\,$ in the range of $\,f\,$ In other words, the composite function $\,f\circ f^{-1}\,$ is the identity function on $\,\text{ran}(f)\,$. ## Input/Output Roles Reversed for $\,f\,$ and $\,f^{-1}\,$

The input/output roles for a function and its inverse are reversedthe inputs to one are the outputs from the other.
This fact has some nice consequences:

POINTS ON THE GRAPHS OF $\,f\,$ and $\,f^{-1}\,$ HAVE THEIR COORDINATES SWITCHED:

$\,(a,b)\,$ is on the graph of $\,f\,$     if and only if     $\,(b,a)\,$ is on the graph of $\,f^{-1}\,$

DOMAIN AND RANGE OF $\,f^{-1}\,$:

• the inputs to $\,f^{-1}\,$ are the outputs from $\,f\,$:   that is, $\,\text{dom}(f^{-1}) = \text{ran}(f)$
• the outputs from $\,f^{-1}\,$ are the inputs to $\,f\,$:   that is, $\,\text{ran}(f^{-1}) = \text{dom}(f)$

## SUMMARY: PROPERTIES OF INVERSE FUNCTIONS

For your convenience, the properties of inverse functions discussed in this and earlier exercises are summarized below.

PROPERTIES OF INVERSE FUNCTIONS
A function $\,g\,$ has an inverse   if and only if   $\,g\,$ is a one-to-one function.

Let $\,f\,$ be a one-to-one function.
The inverse of $\,f\,$ is denoted by $\,f^{-1}\,$ and read aloud as ‘$\,f\,$ inverse’.

The functions $\,f\,$ and $\,f^{-1}\,$ satisfy the following properties:
• $\,f^{-1}\,$ is also a one-to-one function
• the inverse of $\,f^{-1}\,$ is $\,f\,$
• $\,f(x) = y\,$   if and only if   $\,f^{-1}(y) = x\,$
• $\,f^{-1}(f(x)) = x\,$   for all   $\,x\in\text{dom}(f)\,$
• $\,f\bigl(f^{-1}(y)\bigr) = y\,$   for all   $\,y\in\text{ran}(f)\,$
• $(a,b)$ is on the graph of $\,f\,$   if and only if   $\,(b,a)\,$ is on the graph of $\,f^{-1}\,$
• $\text{dom}(f^{-1}) = \text{ran}(f)\,$   and   $\text{ran}(f^{-1}) = \text{dom}(f)\,$
Master the ideas from this section
(when there's only one $\,x\,$ in the formula)