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/
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