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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 
Associated Topics:
High School Functions

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