Why Are Functions Important?

Date: 02/03/2005 at 10:52:56
From: Pardue
Subject: Functions

Why does it matter if something is a function or not?

Date: 02/03/2005 at 22:56:16
From: Doctor Peterson
Subject: Re: Functions

Hi, Pardue.

That depends on the context, of course.  I suppose you're asking why
we teach how to determine whether a relation is a function or not.  
For some ideas on why the concept of a function is important, see

  Why Do We Have Functions?
    http://mathforum.org/library/drmath/view/62559.html 

As to why we would want to determine whether something is a function,
consider the definition of a function.  First, if it is a function,
then we can calculate its value given the input; there is only one
answer.  If it is not a function, then we can't know which answer to
choose.  That's certainly worth knowing.  For example, we know that 
the square root would not be a function if we allowed it to mean 
either root, so we choose to define the radical symbol as representing
 only the principal root.  That decision was made essentially so that 
we would have a function, and expressions involving it would not be
ambiguous.  We want a symbol to have only one meaning, where possible.  
It's also worth knowing that when we work with complex numbers, the
square root is NOT a function; there is no way to consistently define
the square root as having only one value.  Knowing that can help us
avoid some common mistakes.

Similarly, in certain fields of math we define functions in roundabout
ways, and have to show that it is "well-defined", meaning that it
really has just one value that is not dependent on how we define it.  
As an example, you might define a function on rational numbers based
on the numerator and denominator of the fraction, and you have to make
sure that you will get the same answer regardless of which equivalent
fraction you use to do the calculation.  That is an example of
determining whether something is a function, and it amounts to asking,
does what I'm doing make sense at all?

Further, there are some things we can do only with functions-- 
differentiating and integrating in calculus come to mind, and it is my
understanding that the concept of function was invented primarily so
that we could talk about what those operations do (change one function
into another).  We have to know we have a function in order to do 
those things.

Then, too, the fact that a relation is a function is just one more
thing to know about it, which can help in graphing it or otherwise
working with it.

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


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