Domain and Range

Date: 08/25/97 at 12:50:17
From: Jenni
Subject: Pre-Cal

Hi. I'm a junior in high school and I need help with my pre-cal.

I don't understand what we are doing  in class. Can you explain about 
Domain and Range? I don't get it. I also have problems with Piecewise 
Function, Notation, Boolean Test, Transformation on Fraction, 
Operation on Function, Compositions of functions, and domain of 
composite.

Thanks.

Date: 08/25/97 at 13:36:18
From: Doctor Ken
Subject: Re: Pre-Cal

Hi Jenni -

The basic concept of domain and range is pretty simple.  Here are the 
facts:

  - The domain of a function is the set of all the stuff you can 
    plug into the function.

  - The range of a function is the set of all the stuff you can 
    get out of the function.

Let's do an example: f(x) = x^2   (that's x squared)

What's the domain?  Well, you can plug any old real number you want 
into this function: I can square 4, or -7, 1.01738, or whatever, and 
the world doesn't blow up.

What's the range? Well, let's think about it.  If I plug any number 
into this function, am I ever going to be able to get a negative 
number out of it? Nope! (Unless you're dealing with imaginary numbers, 
and I bet you're not!)  So it looks like the range of this function is 
the set of all non-negative numbers (the positive numbers plus zero).  
And in fact, that's the right answer.


Let's try another example: f(x) = 1/(x-5). 
 
What numbers can we plug into this function?  Well, it looks like any 
number will do. But wait! What happens if we plug in 5?  We divide by 
zero, and the world blows up. So let's not plug in 5. Since any other 
number is fine, the domain of this function is: all real numbers 
except 5.

What's the range?  I'll leave that one to you.


Here's an anonymous tip. In general, when you're trying to find the 
domain of a function, there are two things you should look out for.  
They are: 

   (1) Look for potential division by zero.
   (2) Look for places where you might take the square root 
       of a negative number.

Since these are the two easiest ways to make the world blow up, these 
are the things you really have to watch out for.


For your other questions, you can look around in the calculus and 
pre-calculus pages in our archives. Two good pages to start looking 
at are

   High School Calculus
   http://mathforum.org/library/drmath/sets/high_calculus.html   

   High School Functions
   http://mathforum.org/library/drmath/sets/high_functions.html   

There's lots of good stuff there.  Good luck with it!

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/