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/
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