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Topic: [ap-calculus] Piecewise Functions
Replies: 3   Last Post: Aug 21, 2012 1:22 PM

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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
RE:[ap-calculus] Piecewise Functions
Posted: Aug 21, 2012 1:22 PM
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Steven Becker wrote:

> It seems to me that the redundancy of it is what would
> make it NOT a function since you would be defining the
> same point twice. I would explain it to my students this
> way: by definition, for a function to exist, each x-value
> can only be used once and each must also map to only a
> single y-value (i.e. pass the vertical line test).
> If you repeat a point in the domain (even if it is the
> same coordinate point), then by definition you are using
> that x-value more than once, which in turn defies the
> definition. Hopefully this helps.

Unless I'm misunderstanding you, these comments do not
seem to be relevant to any definition of function that
I know of. A function is simply a collection of ordered
pairs that satisfies "the vertical line test". Exactly
how you convey to someone which ordered pairs belong to
the collection, and which ordered pairs do not belong to
the collection is not relevant, at least in so far as
whether the collection represents a function. (It may be
relevant to a reader trying to understand what the person
is conveying, however.)

I understand "f(x) = -2x + 4 on (1,2] and 0 on [2,3)"
to mean that we're to consider the union of the collection
of ordered pairs described by "f(x) = -2x + 4 on (1,2]"
with the collection of ordered pairs described by
"f(x) = 0 on [2,3)". The union of these two collections
is certainly a collection of ordered pairs that satisfies
the vertical line test.

Moreover, descriptions of functions like this (but in other
settings) are used all the time in mathematics. For example,
this occurs when a function is defined on a set of equivalence
classes by defining its action on elements of the equivalence
classes, one of the most common constructions in mathematics
(e.g. consider how ubiquitous it is to show that a function
is "well defined" in most any mathematical endeavor past
the first two undergraduate years of math). There's even
a well known name for situations like this in beginning
topology courses -- the gluing lemma. It shows up in (graduate
level) complex variables courses when analytic continuation
is discussed. It is fundamental in the definition of a
differentiable manifold, namely in the conditions that have
to be satisfied when the domains of the charts overlap.

This said, I consider the student's description something
like writing 4/8 for 1/2 (or maybe 2.3/4.6 for 1/2), or sqrt(25)
for 5, or {1, 2, 2, 3} for the set {1, 2, 3}. It's a correct
description, but perhaps not the most efficient description
or the most useful description.

Dave L. Renfro
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