Function Tests

Date: 02/19/97 at 17:08:59
From: Cindy Smith
Subject: Re:  Functions

I am currently studying functions and am not finding them terribly 
difficult.  However, I do have a question about the reasoning behind 
the nature of functions.  I understand the rules quite well that say, 
in order to be a function, a relation must pass the vertical line 
test.  I also understand that for a function to be one-to-one, it must 
pass the horizontal line test and only functions that pass the 
horizontal line test can have inverses.  

What I don't understand is the reasoning behind the rules.  My teacher 
said this is simply the definition of a function so I shouldn't worry 
about it.  Can you explain to me why it is necessary for every domain 
value to have one range value, but one range value can have more than 
one domain value?  I don't understand why these definitions were made 
for functions.  It just doesn't make intuitive sense to me.  

Perhaps you can tell me what types of problems mathematicians use 
functions to solve that make these definitions necessary.  The 
examples in the book don't really answer this question.  That y is a 
function of x seems to be a law of physics in the math books I've 
consulted.  Please don't tell me I don't know enough math to 
understand the answer; or, if you do, tell me what math I need to 
learn to understand it, at the very least, and I will look it up in 
the library.

Thanks for your time,
Cindy Smith

Date: 02/19/97 at 18:58:34
From: Doctor Ceeks
Subject: Re:  Functions

Hi,

The reason why functions must pass the vertical line test is, indeed,
part of the definition of a function.  But it is something worth
thinking about.

The notion of function is certainly one of the most important notions
in all of mathematics and has its origins in some very concrete 
examples.

Consider this situation: every item in a store has a cost, but no
item has two different prices attached to it.  When you ask: "what
is the price of this object?" you can think of it as asking for
the value of a certain function.  This function is a function whose
domain is the items in the store, and whose range is nonnegative 
numbers (usually, the store doesn't pay you when you buy something, 
which is why I'm saying nonnegative numbers!). You can actually say, 
let P be the function which gives the price of the object.  Then 
P(soap) = 1.29, P(cereal) = 2.29, P(toothbrush) = 2.25 and so on.

The "vertical line test," in the context of the above example, says
that no item has two or more prices (or, in other words, every item
has a unique price).

The "horizontal line test," in the context of the above example, asks
whether, for a given price, there can be more than one item.  That is,
can different items have the same price?  And the answer is, sure they
can.  It wouldn't surprise me at all if the price of a sponge and
the price of a box of tea bags were the same.

The notion of function grew into its important place because people
began recognizing functions everywhere.

For example, in physics, they sometimes ask about the height of
a falling ball as a function of time.

In this example, the "vertical line test" asks whether at a given
time, a ball can be in two different places.  This is impossible!
The "horizontal line test" asks whether the ball can be at a certain
height at different times. If you think of a basketball going up,
then down, you can see that yes, it can have the same height at two
different times.

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