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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

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):

2x
-----  =  y.
(4-x)

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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra

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