Algebraically Equivalent Functions

Date: 06/27/2002 at 12:18:32
From: Jim van Scoyoc
Subject: Algebraically equivalent functions


I'm now officially taking a calculus course, so my involvement has 
gone beyond casual interest.

Anyway, here's my question.  Both my instructor and my textbook have 
presented functions such as

                    x^3-x^2
                     -----
                      x-1,

with the warning that they are undefined if x should be set so that 
the denominator is zero.  Yet, sometimes such expressions can be 
simplified so as to eliminate the denominator, or at least the term 
that includes 'x'.  So the expression above can be re-written as

                    x^2(x - 1)
                     ---------
                        (x -1),

which of course simplifies to

                           x^2.

Yet the original function is still considered to be undefined at 
x = 1.  My instructor and textbook obviously can't be wrong, but why 
do functions work this way?  Isn't it counterintuitive?

Date: 06/28/2002 at 16:13:32
From: Doctor Peterson
Subject: Re: Algebraically equivalent functions

Hi, Jim.

No, it's not counterintuitive, just literal. A function is defined as 
being exactly what we say it is, NOT whatever we can simplify it to.

The function was defined by telling you to subtract the square of x 
from the cube of x, and divide by one less than x. You give it the 
number 0, and it tries to divide 0 by 0 and says it can't do it. So 
the function is undefined at x=0.

You might think of a function as a computer program. It does just 
what you tell it to, with no intelligence to say "well, I can 
simplify it to x^2, so it OUGHT to be 0, and I'll say that instead". 
Some programs try to do that sort of thing, correcting our spelling 
or selecting the part of a text it thinks we probably want, and too 
often it's not what we really intended. We want our functions, like 
our computers, just to do what they are told without trying to add 
their intelligence to ours. Mathematicians are particularly careful 
to say what they mean, and a function that filled in gaps without 
being told to would be very hard to manage!

So you are smarter than a function, and can MODIFY the function by 
simplifying it, which in effect fills in the hole. That results in a 
different function that is nicer, but that doesn't mean the original 
function should be nice and do this without being told.

Does that make sense?

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

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