Finding the Domain of a Function

Date: 09/16/97 at 23:00:28
From: Shelly Breeding
Subject: Finding the domain of a function

I had this problem on a quiz and I have no idea how to figure out the 
domain of this function:

   f(x) = (4+x)/(x^2-9)

Can you please show me how to figure out the domain, maybe by graphing 
it?

Date: 11/10/97 at 11:45:53
From: Doctor Sonya
Subject: Re: Finding the domain of a function

Hi Shelly,

The domain of a function is the set of numbers that you can plug into 
the function and get out something that makes sense. This is also 
called the set of numbers for which this function is defined. In this 
case it is the set of all x, such that f(x) is a number. 

One way to try to find the domain is to try a couple of numbers just 
to get an idea of what might work.

(1) If we plug in 4, we get f(4) = (4+4)/(4^2 - 9) = 8/7.
(2) If we plug in 0, we get f(0) = (4+0)/(0^2 - 9) = 4/-9 = -(4/9)
(3) If we plug in 3, we get f(3) = (4+3)/(9-9) = 7/0 = Oops!

I'm sure you've been told in class that you can't divide by 0, so 7/0 
is an answer that doesn't make sense, and one would say that f(x) is 
not defined at x = 3. You would find that the same holds for x = -3.

Okay, so far we think that the domain of f(x) is all numbers except 
3 and -3. But how do we know this is correst? In principle you would 
have to try every real number other than these two to be sure. In 
practice it is much easier. You just need to ask yourself a question: 
Which is the only way this function can be undefined?  

You know that you always get a fraction out of this function. (If you 
get a whole number n, you can think of it as the fraction n/1.) The 
only fractions that are undefined are those with 0 in their 
denominators. 

We have a zero in the denominator only when:

    x^2 - 9 = 0

or only when 

    x = 3 or x = -3
 
This means that the function is undefined only at these two numbers, 
so its domain is all real numbers but 3 and -3.

I hope that this has helped. 

-Doctors Marko and Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/