Unraveling an Inverse Function

Date: 11/30/2001 at 00:58:00
From: Trevor Lyons
Subject: Inverse functions

I had this question on a test and got it wrong.

f(x) = -5x-2
        ----
        -x+1


the answer is y = -x-2
                  ----
                  -x+5  restriction cannot equal 5.

I have no clue how to work this out, and would really like to know 
how to do it.

Thanks, 
Trevor

Date: 11/30/2001 at 01:18:18
From: Doctor Schwa
Subject: Re: Inverse functions

In other words, y = (-5x - 2) / (-x + 1).

Then the inverse function is what "undoes" that; in other words, the 
opposite, turning y into x instead of x into y.

So, the common way to find it is to switch x and y with each other in 
order to start "unraveling" the function:

   x = (-5y - 2) / (-y + 1).

Now to solve for y, multiply both sides by (-y + 1) to get

   x (-y + 1) = -5y - 2

Distribute,

   -xy + x = -5y - 2

Collect all the y's on one side, and the rest of the terms on the 
other:

   -xy + 5y = -x - 2

Factor out the y,

   (-x + 5)y = -x - 2

and finally divide,

   y = (-x - 2) / (-x + 5)

so

    -1       -x - 2
   f  (x) = --------
             -x + 5

The restriction on x is that the denominator -x + 5 cannot equal zero,
so x cannot equal 5. In other words, the domain of the inverse is all 
real numbers except 5.

I hope that helps clear things up. If not, feel free to write back.

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