Function Machine

Date: 10/26/96 at 19:1:36
From: Brock Johnson
Subject: Domain of a Function

I am working on a chapter entitled "Functions and Limits."  I know 
that the set of all first members is the domain and the set of all 
second members is the range. I don't understand how you find the 
domain or range of a function.

EX.  Find the domain the function f(x)= square root of (4 - x) 

Any help would be appreciated.

Date: 10/29/96 at 1:33:8
From: Doctor Scott
Subject: Re: Domain of a Function

Hi Brock!

Your definitions of domain and range are great when the function is
described using ordered pairs.  So, if the function is f = {(1,2), 
(4,7), (5,8)}, the domain is {1, 4, 5} and the range is {2, 7, 8}.

Another way to think about a function is as a machine.  Functions are 
machines which assign to every INPUT value and OUTPUT value:

                               _____\      /______
                              |                   |
                              |                   |
                              |f(x) = sqrt(4 - x) |
                              |                   |
                              |__________       __|
                                         |     |

Now, the machine operates as follows: put a value in (say 0) and it 
produces an output value (2).  Wow!  Our machine has just produced an
ordered pair (0, 2) for this function! We could, of course, continue 
forever, put every conceivable number in, wait for the machine to 
produce an output value, and then list the domain and range.  That 
might take a while :)  So, there has to be a better way!

Well, consider that the values we put in are what you called the 
"first member" and the values that come out are the "second members".  
Now, think for a minute. Are there any values which we COULD NOT put 
into the machine because they would cause the function to be 
undefined? Well, in your example, the answer is yes!  Any number 
greater than 4 would cause the value under the radical to be negative. 
[We know that the solution to 4 - x > 0 is x < 0]  So, the domain is 
{x | x <= 4}. To determine the domain, we need to think about the 
values which might cause the function to be undefined, and ELIMINATE 
those from the domain.

Now, to determine the range, again, think about the values which can 
be produced by the function machine. In this case, any value greater 
than 0 can be produced. Range is most easily determined by looking at 
a graph of the function - and considering those "y" values which lie 
on the function.

Hope this helps!  Thinking about functions as machines helps in
understanding a lot of the ideas you will study about functions!

-Doctor Scott,  The Math Forum
 Check out our web site!