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RE:[apcalculus] Piecewise Functions
Posted:
Aug 21, 2012 1:22 PM


Steven Becker wrote:
http://mathforum.org/kb/message.jspa?messageID=7871538
> 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 xvalue > can only be used once and each must also map to only a > single yvalue (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 xvalue 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 ==== Course related websites: http://apcentral.collegeboard.com/calculusab http://apcentral.collegeboard.com/calculusbc  To search the list archives for previous posts go to http://lyris.collegeboard.com/read/?forum=apcalculus To unsubscribe click here: http://lyris.collegeboard.com/read/my_forums/ To change your subscription address or other settings click here: http://lyris.collegeboard.com/read/my_account/edit



