Differentiating Variables from Functions

Date: 02/03/2004 at 19:11:55
From: Kathy
Subject: Function Notation

What letters are used to note/name a function?

We've been learning about variables and finally thought that we had 
them figured out.  Now we start learning about functions today and 
they too use letters to show a function.  The kids are asking how in 
the world they are supposed to know when a letter stands for a
function and when the letter is a variable? 

This is one of those things where I just always did it that way and
can't explain why. 

I thought all functions were noted with an f as in f(x), or "f of x",
or "for function f, what is the value of y every time you enter x".   

I haven't had any calculus and wasn't aware that a function could be
written g(x) or b(x).  The kids really want to know how they can 
differentiate variables and function notations. 

Date: 02/03/2004 at 23:43:51
From: Doctor Peterson
Subject: Re: Function Notation

Hi, Kathy.

Any letter can be used for a function, though we tend to use f and g 
when we are talking about generic functions or temporary ones, just 
out of habit, as we tend to use x as a variable when there's no 
reason to give it another name.  Functions that have special names 
usually are NOT called f, to avoid confusion!  For example, Greek 
letters are used for the phi function in number theory and the gamma 
function in analysis.

It can, I suppose, be tricky to distinguish a function from a variable
if you were to see an equation with no context, like

  r(t) = 2s(t) + t(t+1)

(I just made that up to make it hard!)

I know, because of what I meant when I wrote that, that r and s are 
functions and t is a variable; but they could all be variables and it 
would still make sense.  (It wouldn't make sense if t were a function.)

You should also be aware that we can use a variable as a function: if 
we've defined y in terms of x, we can write "y(x)" to mean the value 
of y for a particular x, while still just writing y when we don't 
need to emphasize that relationship.

So what makes the difference between a function and a variable? 
Definitions!  When we write anything in algebra, we have to first 
define our variables and functions.  We tell the reader what things 
mean before we use them. 

Ultimately, that's all that stands between comprehension and chaos.
And that's why I constantly emphasize the need to clearly state the
definitions of variables at the start of a student's work on an
algebra problem. 

You'll see the same thing in legal papers: when you want to use
general terms (like "plaintiff" or "Company") with specific meanings,
you have to say what you mean by them before you start writing.

Pretty simple, isn't it?  Communication 101.

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

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

Date: 02/04/2004 at 08:51:19
From: Kathy
Subject: Thank you (Function Notation)

Oh my goodness!  Now I understand :-) 

Thank you so much for taking the time to answer thoroughly.  Being the
teacher in the class, I felt really silly writing to you, but I've
always told my students that the only truly stupid questions are the
ones you want/need answers to and do not ask!  I appreciate your
efforts on my behalf as I help my young students answer their
'burning' questions.

Date: 02/04/2004 at 09:09:21
From: Doctor Peterson
Subject: Re: Thank you (Function Notation)

Hi, Kathy.

I like this kind of question more than anything else: the ones nobody 
stops to think about, or dares to ask, because they seem too simple! 
It's a great way to stretch my mind.  I'm glad my rambling, until I 
realized how simple the answer was, didn't confuse you.

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