Inverses

Date: 06/05/2001 at 16:10:28
From: Anonymous
Subject: Inverses

What is an inverse?

Date: 06/05/2001 at 17:07:46
From: Doctor Peterson
Subject: Re: Inverses

Hi,

The word "inverse" is used in many different settings, so you may have 
to be more specific about what kind of inverse you have in mind. But 
generally, the word means something that "undoes" something else.

For example, the "additive inverse" of a number is the number you can 
add to it to get zero - in other words, the negative of the number. It 
is called the inverse because adding it undoes addition of the 
original number; if I know what x + 3 is, I can add the additive 
inverse -3 and get the value of x.

Similarly, the multiplicative inverse of a number is the number you 
multiply by to get 1; this is the reciprocal. If I know 3 times x, I 
can multiply by 1/3, the multiplicative inverse of 3, to find x.

We also talk about the inverse of a function. Because x^-1 (x to the 
negative first power) is the multiplicative inverse, we call the 
inverse of the function f, f^-1. (This is a little dangerous, but we 
usually know it doesn't mean the reciprocal of the function.) This is 
the function that undoes what f does:

    f^-1(f(x)) = x for all x

For example, the inverse of the square

    f(x) = x^2, for x >= 0

is the square root

    f^-1(x) = sqrt(x), for x >= 0

because the square root of the square of a (positive) number is the 
number itself.

There are other kinds of inverses, but these are the first you 
encounter.

If you want more details, go to the Dr. Math search page and enter, 
say, "inverse function". Also, check out the Number Glossary in our 
FAQ for my first two examples. Or, write back and tell me more about 
what you need to know.

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