Associated Topics || Dr. Math Home || Search Dr. Math

### The Inverse of the Absolute Value Function

Date: 12/12/2008 at 17:55:59
From: Ed
Subject: Inverse of abs x

What is the inverse of the function abs(X)?  I know the range must be
restricted if the inverse is to be a function, but what equation would
I give to represent the inverse?

I know how to find inverses of the other families of functions, just
not the absolute value functions.

Date: 12/12/2008 at 23:32:18
From: Doctor Peterson
Subject: Re: Inverse of abs x

Hi, Ed.

You have the absolute value function,

abs(x) = |x|

and you want to find the inverse.  So you write

y = |x|

and swap the variables,

x = |y|

then solve for y.  Given a value of x, what can y be?  Well, take an
example.  If x is 5, then y can be 5 or -5, since the absolute value
of both of them is 5.

In general, the solution is

y = +-x

so that

abs^-1(x) = +-x

where "+-" represents the "plus-or-minus" symbol.  Since the range of
abs(x) is x>=0, the domain of the inverse function is likewise x>=0.

This, of course, is not a function, since for each positive x, we get
two values for y. If you restrict the domain of the absolute value
function to x>=0 to make it one-to-one, the function becomes just
abs(x) = x, and is its own inverse!  If you instead restricted the
domain to x<=0, then abs(x) = -x, and this time the inverse is
abs^-1(x) = -x, with a different domain (x>=0). This reflects the
fact that |x| can be defined piecewise as

|x| =  x if x>=0
-x if x<0

Either "piece" has an inverse.

Note the graphs of our two functions:

abs(x):                       abs^-1(x):
\         |         /                   |         /
\       |       /                     |       /
\     |     /                       |     /
\   |   /                         |   /
\ | /                           | /
----------+---------          ----------+----------
|                             | \
|                             |   \
|                             |     \
|                             |       \
|                             |         \

y = |x|                       y = +-x, x>=0

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
Associated Topics:
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search