What The Heck, Some Math

This has been a record-setting day for this blog (4 posts). Today I did the last of the three “Introduction to Piecewise Functions” lectures. So far I haven’t written out anything like what follows in class and I’m not at all sure I ever will. But it occured to me last night, that as long as we’re taking the whole “sets of ordered pairs” thing seriously, we could consider a piecewise defined function f(x) = \begin{cases} x^2 & if \quad\quad x < 2\\ 3-x & if \quad\quad 2 \le x \le 4 \end{cases}
from a set-theoretical point of view … we’re “really” considering the union of certain sets, which are themselves the intersections of certain others. Here we go.

The first line (in the right-hand side of the definition of f) can be thought of as telling us to consider
\{ (x, y) | y = x^2\} \cap \{ (x,y) | x < 2\}
the intersection of a parabola and the “half-plane” to the left of the vertical line x = 2.

This can be abbreviated further, as \{(x,x^2)\}_{x \in (-\infty, 2)}—the restriction of the squaring function to the (“subdomain”) (-\infty, 2). Using this language, we then have
f = \{(x,x^2)\}_{x \in (-\infty, 2)} \cup \{(x, 3-x)\}_{x\in [2, 4]}.
There it is. I don’t think I’ve ever seen a piecewise function spelled out this way and for some reason I find it sort of charming. Better with the pictures, of course. I’d pay a lot of money to find out how to post nice graphics for free.

  1. brutus12

    Not sure if a “comment(s)” can be a “question(s)” so if not I apologize but if this is okay here’s the question…

    I understand the above such that…
    begin by graphing both functions x(squared) AND 3-x… then highlight the x’s that are less than 2 on the x(squared) graph and also highlight the x’s on the 3-x graph that are greater than or equal to 2 but less than or equal to 4 with such an answer of the domain of x(squared) with the above conditions is (-infinity, 2) and the domain of 3-x with the above conditions is [2,4]

    Is this correct?

  2. not only *allowed* but *encouraged*:
    questions might well be the best *kind* of comment!

    i think you’re seeing the picture pretty clearly …

    begin by graphing both functions x(squared) AND 3-x… then highlight the x’s that are less than 2 on the x(squared) graph and also highlight the x’s on the 3-x graph that are greater than or equal to 2 but less than or equal to 4

    thus far i’m fairly sure i understand you

    with such an answer of the domain of x(squared) with the above conditions is (-infinity, 2) and the domain of 3-x with the above conditions is [2,4]

    this bit isn’t quite so clear.

    i’m actually thinking of taking the x^2
    (internet standard for “x-squared” …
    that this looks like the calculator code
    makes it easy to remember … that it’s
    also the “TeX” [math typesetting program]
    is a nice bonus …), taking the x^2, i say,
    and “highlighting” the bits with x’s
    less than two (i usually think of this as
    “throwing away” the part of the x^2 curve
    that’s *not* in the half-plane x >= 2
    [>= here denotes “greater than or equal to”;
    this *isn’t* standard … but is pretty common;
    among the math-heads, \ge is probably more so …
    DOLLARSIGN latex \ge DOLLARSIGN produces \ge].
    this “highlighted” piece of the x^2 curve is
    the restriction of the squaring function
    to the interval (-infty, 2): if we consider
    s(x) = x^2 to be the “usual” squaring function
    (i.e., the “whole thing”, with {\mathcal D}(f) = {\Bbb R}),
    the symbol s|_{(-\infty, 2)} for this
    restricted function is pretty common; since i’m stressing
    the “ordered pair” point-of-view, i’ve referred to
    \{(x,x^2)\}_{(-\infty, 2)}.

    anyhow, next we find — *draw*, really, since
    in actual exercises, we’re typically constructing
    *pictures* of the sets in question — the restriction
    of the “subtract from three” function to [-2,4].
    forming the union of the two restricted functions
    then amounts to drawing the curve and the line
    segment and sticking a label on it (y = f(x) here).

  3. part of the point i was making at such great length
    a moment ago is that, from the point of view
    taken in this post
    , we’re constructing the
    individual restrictions-of-sets *first* (by forming
    the *intersections* of certain sets), and only then
    putting ’em together into a single function
    (which is the *union* of the restrictions).

    in working out actual textbook problems, it’s common
    to graph both “full” functions on a single co-ordinate frame
    and then, yes indeed, *highlight* certain parts
    of each of the relevant pieces … i’ve recently called this
    part of the procedure the “first draft”, to be followed
    by a new picture consisting only of the highlighted pieces
    (the “final draft”). there’s sure nothing wrong with this.

    the restrict-individual-sets-first approach i’ve described
    sure isn’t offered here as a better way to “get the answer”;
    rather, i’m focussed on how to *describe* the answer
    in the set-theoretic language …

    it’s a lot of trouble, to be sure. but the advantage
    is that we’re able to say, in at least some sense,
    *exactly* what we’re talking about …
    “union” and “intersection” are the kind of thing that,
    for example, can be explained *to a computer*
    in some already-existing format, whereas a concept
    like “highlighting”, easily explained to a human
    (that cares; motivation must of course be assumed),
    would require some special-purpose coding to computerize.

    the goal of all of our mathematics somehow seems
    to get all the *thinking* done once-and-for-all
    (for a given problem-type, say … then think
    about something *else*); having done this,
    we can then simply *calculate* (or let a robot do it).
    a well-designed *notation* is one of the most powerful
    tools in this endeavor.

  4. brutus12

    i think i understand this now :-) the problem in class was tough, but i understood it so maybe that means i did good on the quiz lol

  5. W1ng5Up

    While we are on the subject of “questions . . . encouraged” could you offer a prediction for text sections to be covered the next two – three weeks? I want to keep up-to-speed if not forging ahead, but it would be helpful to plan around work schedules (although math is my be-all and end-all for this quarter, income needs loom on the horizon lol). No expectation for assignments “written in stone,” a moving target is welcome. Thanks!

  6. thanks for the noodge, Wing5Up …
    there’s a day-to-day breakdown
    created by the course co-ordinators.
    i’d’ve published it in the syllabus
    but for some reason they ask us not to.
    anyhow, in principle, *i*’ve got it
    (though, and this won’t surprise you,
    i can’t put my hand on it right *now* …).
    and it’s important that i check it from time to time.
    so i’ll be sure to check it publicly–
    i.e., make an in-class announcement
    of the game plan for the next couple weeks.

    do please remind me of this during class
    if for some reason i seem to’ve forgotten …

    meanwhile, more or less obviously i hope,
    we’ll be starting to look at quadratic equations
    next (with careful reference to how the theory
    of *transformations*–our latest topic–can be
    made to shine its light on “y = x^2” and its relatives).

    “forging ahead” is of course a wonderful idea;
    in the current context, this might very well
    have the somewhat-paradoxical flavor of *looking back*:
    one way to be prepared for what’s coming is
    to be *very confident* with (a) factoring polynomials
    and (b) working in {\Bbb C} (the complex #’s)–
    to name only two topics introduced in earlier courses
    (but here to be explored in much greater depth).
    how i’ve escaped making an enormous fuss
    about these topics (several times!) in class so far
    i’ll never know … they’re coming up fast …

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