Are All Functions Equations?

Date: 07/16/2001 at 16:42:31
From: Maureen Hamilton
Subject: Are ALL functions equations?

I know that any equation that is written y = f(x) is considered a 
function, since when it is graphed it passes the vertical line test.  

I also know that I can draw a horizontal wiggly curvy line that is 
also a function since it also passes the vertical line test. However, 
this second function cannot be defined by a single equation. I suppose 
it could be considered a multi-piece function, and around specific 
turning points it could be defined by an equation.  

I also know that a table of (x,y) values can define a function as long 
as every x has only 1 y associated with it. However, if I pick quite 
random values for my x's and y's, once again I cannot define this 
table with an equation. Also, since my x's would not be continuous - 
assume all x's are integers - would I still have a function since the 
vertical line test might in fact not touch a point at all? 

I have been searching for the definitive answer to this question for 
many years.

Date: 07/16/2001 at 22:58:02
From: Doctor Peterson
Subject: Re: Are ALL functions equations?

Hi, Maureen.

My first impression is that the following answer may make a start at 
answering your question:

   Functions and Equations
   http://mathforum.org/dr.math/problems/abby.02.10.01.html   

But your question is a little more detailed, so I'll try to answer you 
directly as well.

First, we have to define clearly what we mean by "function" and 
"equation." A function is not an equation; rather, it is a relation. 
It need not be written as an equation at all, but can be any relation 
between two variables such that for every value of the independent 
variable within the domain of the function, there is one value for the 
dependent variable. An equation is merely one way to express a 
relation.

It sounds as if you are thinking of an equation as having to be 
expressed in a certain way - perhaps always having a polynomial or 
rational expression on each side. But if you think carefully, you will 
recognize that many special functions have been defined just because 
there was no other way to express them except in functional notation. 
For example, we have the trig functions sin(x) and so on; in your 
view, am I writing an equation if I say

    y = sin(x) ?

I imagine so. This can't be expressed in terms of rational functions, 
but by allowing trig functions to be included in equations, we have 
written an equation. Now consider the absolute value:

    y = |x|

This could just as well be written as a function:

    y = abs(x)

But apart from these notations, we would have to define this equation 
piecewise:

    y = x  for x >= 0
        -x for x < 0

Does that make this any less an equation? By defining the absolute 
value function, we have stitched together two linear functions into a 
single expression so that it looks like a single equation. But if 
there is any other function you have in mind that can't be written as 
a single equation USING ESTABLISHED NOTATIONS AND FUNCTIONS, why can't 
we just define a new function notation for it and write it as an 
equation?

So I would say, on one hand, that there certainly are functions that 
can't be expressed as an equation of a FAMILIAR FORM, but there is 
nothing essentially different about such a function; it's just a 
question of what we consider familiar. Perhaps if you can define 
exactly what you mean by "equation," I could give a more specific 
answer.

Now, if you start with a table listing a finite set of x-y pairs, then 
it is not hard to define a function - in fact, a polynomial function - 
that fits it. If some x values are missing from what appears to be the 
natural domain, we can just exclude those values when we define the 
domain. For any function to be defined, we must state (or imply) its 
domain, so this is no real restriction. A function on a discrete 
domain is perfectly legitimate. If you express it in terms of an 
equation, of course, the domain has to be expressed separately if it 
is not the natural domain, so in such a case the function can't be 
fully expressed by ONLY an equation.

Perhaps ultimately the answer to your question is to recognize that 
the function concept is a generalization of the equation concept, 
allowing us to talk about "curvy lines" without being able to write a 
specific equation. Not all functions can be expressed in any familiar 
way as an equation, but all functions can be used to make equations.

I've probably missed some aspects of your question, so write back if 
you aren't fully satisfied.

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

Date: 07/17/2001 at 12:50:35
From: Maureen Hamilton
Subject: Re: Are ALL functions equations?

Thanks for your response. You cleared up some of my confusion. I guess 
one of my sub-questions is, what if I have a table of values that 
appear to be random but that do in fact define a function - for 
instance the domain is all the social security numbers in the U.S., 
and the range is the weight of the person having that social security 
number? 

If I knew what all the numbers were and they were defined in a very, 
very long table, could I make life easier for myself by determining an 
equation (polynomial, rational expression, trig, etc.) such that I 
could use this equation to determine the weight of the person having a 
particular social security number f(social security number) = weight 
rather than having to look up the weight in a 
table?

Date: 07/17/2001 at 21:52:56
From: Doctor Peterson
Subject: Re: Are ALL functions equations?

Hi again!

In general, you could theoretically make a polynomial that matches the 
table, but it wouldn't help at all (unless it's not as random as it 
looks). For any n points, you can find a polynomial of degree no more 
than n-1 that passes through those points - so the expression you 
would get would probably be about the same size as the table itself.
A computer, in particular, would handle the table much better than the 
polynomial. And, of course, you would also have to find a way to check 
whether a given number was in fact a valid SSN, and I can't think of 
any nice way to do that without the table.

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

Date: 07/18/2001 at 21:51:41
From: Maureen Hamilton
Subject: Re: Are ALL functions equations?

Thank you so much - you really have in fact answered my question. And 
now quickly... If I have a table with 5 points I assume the polynomial 
would be a degree 4 equation (or less). How do I find this polynomial?  
I know if I have 2 points how I would find the equation of the 
corresponding line.  I am assuming 3 points will give me a parabola, 4 
points a cubic, 5 points a W-shaped parabola, etc. I am unfamiliar 
with the math that would enable me to find the equation of a parabola 
given 3 points (not in a straight line) as well as finding the cubic 
equation defined by 4 points. Is there a simple answer to this 
question or can you point me in the direction of a math 
book with the answer?  

ONCE AGAIN, THANK YOU SO MUCH. Believe me, this has been nagging at me 
for some time.

Maureen

Date: 07/18/2001 at 22:04:33
From: Doctor Peterson
Subject: Re: Are ALL functions equations?

Hi again!

Here's a quick answer:

   Finding Polynomials
   http://mathforum.org/dr.math/problems/elizabeth4.29.97.html   

I found this by searching our archives for the words "polynomial fit 
points" . There are also methods to find a "closest fit" using a 
simmpler polynomial, but that's not what you want. There are probably 
other explanations of this process in the archives if this isn't full 
enough.

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

Date: 07/19/2001 at 13:23:28
From: Maureen Hamilton
Subject: Re: Are ALL functions equations?

Hi,

I just want to say thanks.  I needed to know if a polynomial could be 
arrived at from a table of values, even if it's very complicated.  I 
shall follow your link but for the most part you answered a question 
that has been nagging me for some time.  Thanks again.

Maureen