Re: Undefined function question...
- From: Bill Dubuque <wgd@xxxxxxxxxxxxxxxxxxxx>
- Date: 28 Aug 2005 05:10:04 -0400
"." <.@.com> wrote:
>
> If I have a function: f(x) = X^2 + 10/(X-2)
>
> and X is 2, the second part of the equation is undefined, correct?
> If I want to find the maximum domain and range of the function,
> how do I handle this? Do I just ignore the undefined section
> because of the defined component?
[I'm going to discuss a more general problem, which will reveal
a pretty answer to an unsolved problem I previously posed here]
As stated above the problem is ill-posed since the function is not
rigorously specified. In particular neither the domain nor codomain
of the function is specified. Assuming "standard" denotations it
is probably safe to assume that one intends it to denote a function
over some field, most likely the real or complex number field.
However, it could just as well be interpreted as a function over
the field of five elements Z/5 := the ring of integers modulo 5.
If it denotes a possibly partial function then it is undefined
for x = 2. However, if it denotes a rational function then it's
equal to the function f(x) = x^2 because 10/(x-2) = 0/(x-2) = 0
as rational functions over Z/5, simply because in every ring
0*a = 0. Recall rational "functions" aren't generally functions
but rather are elements of the fraction field of polynomials
(*formal* polynomials, not polynomial *functions*).
Sometimes one knows *a priori* that the solution of a problem
must be a (formal) polynomial, so that one can exclude solutions
that are not total or have singularities. For example, if one
knows that the function f(x) has polynomial form and that it
satisfies x f(x) = x then the unique solution is f(x) = 1
because polynomial rings (over a domain) are themselves domains
so satisfy the cancellation law, so one may simply cancel x above;
in other words the rational function x/x = 1. While this may
seem like a trivial observation, it can prove quite powerful in
less trivial contexts. E.g. in 1999 I posed this problem [1]:
Let A be an n by n matrix over a commutative ring.
n-1
Show |A~| = |A| , A~ = adjoint A, |A| = det A
T
Note: since A A~ = |A| I, taking determinants
n
yields |A| |A~| = |A|
but |A| can't be "naively" canceled if it is 0 (or a zero-divisor).
The solution is just as trivial as above because we know *a priori*
that both |A| and |A~| have polynomial form (in the entries of A)
so that is it valid to cancel |A|. Surprisingly, no one on sci.math
noticed this simple solution - the only posted solution, by Ted Hwa,
employed a topological (density) argument. As Arturo Magidin might
say, that is swatting flies with cannons.
Enough digression on semantics vs. syntax, function vs. form.
Thus, in summary, not only do we need to know the denotation of
the operations in your expression, but we may also need to know
the denotation of the term "function", since it might denote a
purely *formal* object in some ring rather than a true *function*.
Hopefully I haven't been too *dense* above (pun intended!).
--Bill Dubuque
[1] http://google.com/groups?selm=y8zogk3m92r.fsf@xxxxxxxxxxxxxxxx
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