Function Its Own Inverse

Date: 01/02/98 at 22:40:07
From: Jaime
Subject: Inverse function

Show that f(x)= __X__  Is it's own inverse.
                 X-1

I have tried substituting x in for y and y in for x, but I always get 
stuck.

Date: 01/07/98 at 10:58:27
From: Doctor Jaffee
Subject: Re: Inverse function

Hi Jaime,

It sounds to me as if you are off to a good start. Making the 
substitutions that you suggested will work if you then isolate the y.  
The result should be y = x/(x-1), which is exactly the result you need 
to prove that the function is its own inverse.

There is another method you can use, however. If g(x) and f(x) are 
inverse functions, then g(f(x)) and f(g(x)) will both equal x. In this 
particular case, you are trying to show that a function is its own 
inverse, so it is sufficient to demonstrate that f(f(x)) = x. Give it 
a try. If you can handle complex fractions, you should be successful.

Also, try graphing y = x/(x-1) and y = x on the same grid. You should 
notice that y = x/(x-1) is symmetrical around the line y = x. That is 
another way to show that the function is its own inverse.

Can you figure out why these suggestions work?

-Doctor Jaffee,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/