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### Relations versus Functions

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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
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```
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
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
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/
```
Associated Topics:
High School Basic Algebra
High School Functions
Middle School Algebra

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