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### The Vertical Line Test for Functions

```Date: 05/06/2004 at 18:07:15
From: Angie
Subject: Functions

I'm having a problem understanding functions.  I understand that for
every "x" there must be only one "y".  But the vertical line test is
what is confusing me.  For example, given these points:

(2,0), (0,2), (1,3), (3,5), (5,3)

domain = 2,0,1,3,5
range  = 0,2,3,5,3

I can see that each "x" has its own "y" but these coordinates don't
make a "straight" vertical line.  Is this a function?

```

```
Date: 05/06/2004 at 23:01:09
From: Doctor Peterson
Subject: Re: Functions

Hi, Angie.

The idea of the vertical line test is not that the points have to form
a vertical line, but that only one point can be on a given line.  In
fact, if they do form a vertical line, it is NOT a function!  That's
really useful only when you are given a graph, where you can just
imagine cutting it with a vertical (up and down) line and seeing that
the line always cuts it in only one point (at most):

vertical
line
|
|           *
|    *           *
*     *                  *
* * |                   *
------+---------------------

If it could ever cut the graph in two points, like this, then it would
not be a function:

vertical
line
|
|       *
*
*   |
|  *    *
*     *           *
* * |
------+---------------------

When you are given a table rather than a graph, you just want to
confirm that no x value appears more than once in the table with
different y-values.

These tests are just two ways to apply the real definition of a
function: it has to take a given number x into one specific number y,
without forcing you to choose one of several.  The vertical line test
does that because any points where the line intersects the graph
correspond to values of the function (y) for the same x (a vertical
line represents one particular vlaue of x).  The table test does the
same thing because two entries in the table with the same x give
different values of the function (y) for that x.

By the way, when you state the domain and range, it's a good idea to
sort the list of values; in this case, that would catch the fact that
you have listed one value twice.  Remember, the range is a set, and a
number can only be an element of a set once.

If you have any further questions, feel free to write back.

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

```
Date: 05/12/2004 at 18:36:01
From: Angie
Subject: Thank you (Functions)

Thanks to you I now get it!  I can now see that the vertical line test
is just a quick check to be sure that there aren't two "y's" for one
"x".  I truly get it now!  Thanks so very much!!

Angie
```
Associated Topics:
High School Functions

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