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Finding the Inverse Function

Date: 04/10/97 at 20:55:44
From: merri brennecke
Subject: Finding the Inverse Function

Find the inverse function of f(x)=4x/(x+2), where x does not = -2. 
State the domain and range of each. 

I am unable to solve for x.  The answer I get is y = 2. When I solve 
for x, they cancel each other out. 

Thank you for your help. Merri

Date: 04/11/97 at 20:34:26
From: Doctor Wallace
Subject: Re: Finding the Inverse Function

Hi Merri!

Recall that inverse functions are symmetrical about the line y = x.  
Since this is true, the way I was taught to find an inverse was to 
swap the x and y variables and solve for y.  This is how it works in 
your problem:

Begin with

         4x                          4y
   y =  -----  and swap to get  x = ----- .
         x+2                         y+2

Now we solve for y.

First we get rid of the denominator on the right by multiplying by 
y+2.  This gives:

   x(y+2) = 4y.

Now, use the distributive property on the left, to get

   xy + 2x = 4y.

Group the y's on the right, to get

   2x = 4y - xy.

Factor the right side:

   2x = y(4-x).

And finally divide both sides by (4-x):

   -----  =  y.

And so there's the inverse!  You have to check to make sure that 
it's a function, and not just a relation.  We have to restrict the 
domain here, as well, or we get a division by zero, so we say x not 
equal to 4.

For the domain and the range, we have to keep the restrictions.  
Since the original could not have -2 in the domain (x values),
the inverse cannot have -2 in the range (y values).  Sure enough, 
if you plug a -2 in for y in the inverse, you get -8 = 0 when you 
simplify it.

I hope this helps.  Thanks for writing!  Don't hesitate to write 
again if you have more questions.

-Doctor Wallace,  The Math Forum
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Associated Topics:
High School Basic Algebra

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