Domain and Natural Domain: What's the Difference?

Date: 01/13/2010 at 19:14:12
From: Ayah
Subject: Domain and natural domain. 

From what I understand, the natural domain of a function is the 
largest set (in the sense of containment) over which the function is 
defined.

The mathematical domain may be restricted to a subset of the natural 
domain.

Does that mean that the domain can be restricted but the natural 
domain cannot?  This is what I find confusing!

Or are the terms interchangeable?  I don't understand the difference.

Date: 01/14/2010 at 10:16:22
From: Doctor Rick
Subject: Re: Domain and natural domain.

Hi, Ayah.

Every function has a domain, the set of (input) values over which it 
is defined.  If I don't state what the domain is, by convention we 
take the domain to be all (real) numbers for which the expression 
defining the function can be evaluated.  We call this the "natural 
domain" of the function.

Let's look at an example.  Take the function f(x) = 1/(1 - x).  This 
function can be evaluated for all x except x = 1, because replacing x 
with 1 means dividing by zero -- an undefined operation.  Therefore, 
1/(1 - x) is not defined for x = 1:

  {x | x != 1}     (the pair of symbols != means "is not equal to")

If I wanted to, I could state a domain explicitly, for instance

  g(x) = 1/(1 - x), for x < 1

In this case, the domain is

  {x | x < 1}

Why?  Because I said so!  It isn't "natural"; I had to declare 
something to make it so.  By stating the domain explicitly, I am 
saying that this function g(x) has no value for any inputs greater 
than or equal to 1, even though I would have no trouble evaluating
1/(1 - x) for, say, x = 2.  I have restricted the domain.

In summary, the natural domain of a function (or rather, of the 
expression used to define the function) is one particular domain -- 
the one we assume when no domain is stated explicitly.

When we do state the domain explicitly, it must be a subset of the 
natural domain.  In my example, I can't say h(x) = 1/(1 - x) for all 
real numbers, because it has no value for x = 1.

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

Date: 01/14/2010 at 10:46:36
From: Ayah
Subject: Thank you (Domain and natural domain. )

Thank you very much.  I really appreciate your response.  You really 
helped me understand it better.  My textbook wasn't very clear, but 
your examples are excellent!!

Thank you very much!