Natural Domain of a Function

Date: 03/12/99 at 12:01:01
From: Tara
Subject: Natural Domain

My teacher has been teaching us about natural domains and other 
related concepts, but I do not understand the whole concept. Could you 
please explain them in detail? He told us that they were the largest 
set of inputs for which something exists. This definition does not 
help me. Could you try to explain it in another way that might make 
more sense to me?

Date: 03/16/99 at 13:15:47
From: Doctor Mike
Subject: Re: Natural Domain

I will start out in what may seem to be a strange way - but stay with 
me. 

Think about what it means to talk about a birth month. We could number 
the months from 1 to 12 so that my birth month would be 7 because I 
was born in July. Abe Lincoln's would be 2 because he was born in 
February. We could express the birth month number as the output of 
function M (has your teacher used the word 'function'?) given the 
input of any person:
   
       M(Dr. Mike) = 7
       M(Lincoln)  = 2 
  
I do not know when you were born, but I do know that 'M(Tara)' is
something - 1 or 2 or . . . or 11 or 12 - even though I do not know 
specifically which number between 1 and 12 it is. What about 
  
       M(President Clinton's cat named Sox) = ?

Again, I do not know what this number actually is, but I do know that 
the President's pet cat was in fact born sometime, so that input makes 
sense for that function. What about

       M(The White House in Washington, D.C.) = ?
       M(Paris, France) = ?
       M(Plymouth Rock) = ?

None of these makes sense, because - unlike living beings - buildings, 
cities, and landmarks are not born, so they cannot be born in a 
particular month. It just does not make sense to talk about their 
birth months. So, the natural domain of the function M is the set of 
things that were born sometime.

I just wanted to start you out with an example of a function that is 
not defined completely in terms of numbers. Now let us go to something 
that is probably more like the examples you have seen. 

Consider the 'square root' function. If your teacher has not yet used 
the word 'function', you can instead say the square root 'rule' or the 
square root 'process'. I will write this in shorthand as a function S:
   
       S(x) = the real number which is the square root of x

So, S(9) = 3, and S(144) = 12, and so on. Certainly S(Plymouth Rock) 
does not even make sense. In addition, expressions like S(-9) do not 
make sense either, when the only kind of numbers you are talking about 
are real numbers (you may learn about complex numbers later). The kind 
of numbers x where S(x) make sense are the real numbers greater than 
or equal to zero. This is the natural domain of the S function.

Let us see about a similar but different function D defined as
  
       D(x) = the real number which is the square root of x - 13
   
You could also write this function as

       D(x) = S(x - 13)
  
using the function S that we talked about above. For this to make 
sense, x - 13 must be greater than or equal to zero, which is 
equivalent to saying that x must be greater than or equal to 13. 
So, the natural domain of D is the set of real numbers >= 13.  

I hope these examples help you to understand what this natural
domain idea is all about.  

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