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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

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/
```
Associated Topics:
High School Functions

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