Relations versus Functions

Date: 10/27/98 at 22:58:00
From: NICOLE
Subject: Functions

Dear Dr. Math,

In Algebra 2, the chapter we are working on now is about functions and 
linear functions. How can you tell whether an equation is a function 
or just a relation? Given the domain and the range, how do you decide 
whether or not a relation is a function? Please help.

Date: 10/28/98 at 09:56:02
From: Doctor White with Doctor Teeple
Subject: Re: Functions

Nicole:

First, all sets of ordered pairs are relations. A function is a 
relation with a certain criterion. The criterion is that for every 
domain value put into the function, the function will create only one 
range value.

We think of domain elements as being values we put into a function to
produce a result. For an example of a function, let's use a ruler. You
may not think of a ruler as a function, but each time you measure an
object, you assign the object a number. So in this case, the elements
in your domain are the objects you measure. The range consists of the
possible numbers you can get when you measure. In the ruler example,
the range is the positive real numbers, since you can't have negative
measurements.

When you indicate your function, to be really clear it is best to
define your domain and your range, in addition to your function. For
example, suppose we measure the length of a pen, the length of a table, 
and the length of a rug. The domain is the set A = {pen, table, rug}. 
Then we would say something like: "Let f be a function from the set A 
to the reals, defined by measuring the object with a ruler." For a more 
mathy example, we could say: "Let f:R->R be defined by f(x) = 5x-2", 
where the notation f:R->R means: the function f from the domain R 
(reals) to the range  R.

One thing we have have to watch out for in functions is to be sure that 
an element in the domain doesn't go to two elements in the range. I'll 
give you an instance using our ruler example. 

Suppose we are trying to figure out the length of the pen. We take the 
ruler and measure 6.3. But then someone else measures and says that he 
thinks the answer is 16. What's going on here? Well, we figure out that 
we measured in inches, and the other person measured in centimeters. So 
who's right? They are both measurements, so by our "function" we are 
both right. That can get pretty confusing, and we need to make sure 
that each element maps to only one element in the range. We can show 
this a couple of ways.

Numerically: Let x = y^2. If x = 1, then y can equal 1 and -1. One 
domain produced two ranges. This is NOT a function.

Graphically: If you sketch the graph of the relation, you can use the 
vertical line function test. Sketch the graph. If you can draw a 
vertical (up and down) line through any x-value, and it hits the graph 
at more than one place then it is NOT a function. If it hits it only 
once then it IS a function.

I hope all of this helps you see this a little better. Please note that 
this is only one of the conditions we have to check to make sure that a 
relation is a function. For more information, please see the following 
entry in our archives:

  Relations on a Set, as Mappings
  http://mathforum.org/dr.math/problems/turner.7.19.96.html   

Come back to see us soon. If you need more information, let us know.   

- Doctors Teeple and White, The Math Forum
  http://mathforum.org/dr.math/