Inverse Functions - Which Statement is True?

Date: 09/20/98 at 21:19:32
From: Faith Smith
Subject: Pre-Calculus-Inverse Functions

Hi, my question is this:

Which statement is true?  Prove it.

 (i) (f composed with g) raised to the (-1) = f raised to the (-1) 
     composed with g raised to the (-1)

(ii) (f composed with g) raised to the (-1) = g raised to the (-1) 
     composed with f raised to the (-1)

Thank you, 
Faith

Date: 09/21/98 at 13:27:19
From: Doctor Peterson
Subject: Re: Pre-Calculus-Inverse Functions

Hi, Faith. I'll just give you a start, so you can have the pleasure of 
finding the answer yourself.

If we have a guess as to the inverse of some function, how can we prove 
that it really is the inverse? We have to show that it satisfies the 
definition of inverse. If we say that F is the inverse of f, we have to 
show that:

   F o f = I  and  f o F = I

where I'm using "o" for composition and I for the identity function.

So for each of your two statements, put the thing on the right for F 
and the thing under the -1 on the left for f in each of these 
equations, and see if it is true. For example, for the first, the 
claim is that:

    (f^-1 o g^-1) is the inverse of (f o g)

so you would want to show that:

    (f^-1 o g^-1) o (f o g) = I

and:

    (f o g) o (f^-1 o g^-1) = I

You have been warned that they aren't both true, so if you get stuck 
proving one, you should try doing the other. You may want to test each 
one first with two simple functions (maybe f = 2x and g = x-1) to see 
if it works. Or maybe you are supposed to know which is right, and 
should just choose that one and prove it.

When you find the answer, you will have shown why, when we solve an 
equation like 2x - 1 = 3, we "undo" the operations that were done to x 
(multiplying and then subtracting) in reverse order (adding and then 
dividing) to get (3+1)/2.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/