Bricks Without Straw

OK. Look. Last week’s rant was exhausting and I don’t want to make a career out of railing against this doggone book. But come on. How the bejabbers am I supposed to talk about inverse functions without a notation for the furshlugginer Identity Function? I mean, seriously.

Consider the set of Real-valued functions of a Real variable, then. Recall that yesterday we defined the composite of f with g by f \circ g (x) = f(g(x)). We (okay, I) failed to mention that the commonest ways to pronounce the lefthand side are “f circle g”, “f composed with g”, and “f composite g”. I’ll have said it in class, though, if only to urge everyone not to pronounce it “fog”… what’s obvious here—but is not obvious in my handwritten work—is that the Circle symbol isn’t the letter O.

Anyhow, those are indeed all common; you pretty much have to become comfortable with at least one of ’em to talk about this stuff at all. Assuming everyone’s done enough practice problems with composite functions to make sense of some calculations, then, we’ve got some tools in hand; let’s see what we can do with ’em.

Yesterday we looked briefly at functions I called a, m, and p—the “add one” function, the “multiply by three” function, and the “raise to the power five” function. At the risk of belaboring a point, I’ll mention right now that the “scarequotes” should not be taken to indicate that there’s anything in the world wrong with calling these functions by these names—indeed, I’d be glad to argue that these are in at least some ways better names than, say, a, m, and p.

Formally, of course, one has a(x) = x+ 1, m(x) = 3x, and p(x) = x^5 (as I put it yesterday), or
a := [x\mapsto x+1]
m := [x\mapsto 3x]
p := [x\mapsto x^5]
(as I only wish I had the nerve to try to pull off in 148 this quarter). The notations all by themselves don’t make the nature of these functions any easier to understand than their verbal descriptions… though for “messier” functions there will come a time when words fail and we have to write things down to keep track… what the notations do give us, though, is something to calculate with.

I also mentioned functions called a^{-1} and m^{-1} yesterday, but failed even to name ’em—”a-inverse” and “m-inverse”—never mind define ’em.

So here we go. Define the identity function, I, by I(x)=x. This function is just as essential to the theory of compositions as the number 0 is for the theory of addition (or the number 1 is for the theory of multiplication): an object that does nothing gives us a simplest possible case… to which every other case can then be compared. If you don’t understand this parable, how will you understand all parables?

Defining inverses is now a simple matter. Suppose f and g (real-valued functions of a real variable, as you will recall) satisfy f\circ g = I and g\circ f = I. We then call g the inverse of f, and write g = f^{-1}. And that’s it; everything else is consequences.

An equation of functions, f\circ g = I, say, means that the functions on its either side have the same domains and that for any “input” value from that domain, they both evaluate to the same “output”: f\circ g (x) = I(x), \forall x\in D, in our example. The symbol D here stands for the Domain in question (some subset of {\Bbb R}, of course), and “\forall“, as usual, means “for all”… hmmm. It seems to be a habit with me to assume familiarity with the \in symbol here but this is very likely a bad habit; this symbol typically denotes “is an element of” (though in this context, I would most likely pronounce it “in”; hovering your mouse over the code will reveal that this is no mere idiosyncracy of mine [ Or not. This feature isn’t working on my equipment at this time. I’ve seen it done. One typically blames the user at this point. It’s not your fault if this doesn’t work]).

Those with a taste for the technical might feel that “really” an equation of functions ought to mean that certain sets of ordered pairs are equal (as sets…”coextensive”, as one sometimes hears it said [but without regard to order; the point to this digression is that, for example, one has \{37, 168\} = \{168, 37\} as sets; the digression itself is offered for the plain fun of it here (since, when working with subsets of {\Bbb R}, it does no harm to assume that they’re given the standard “number-line” ordering; the issue doesn’t even arise)]). I’ve done the calculations, just now (right here at the keypad), and decided not to publish; suffice it to say that they were messy enough to convince me I’d lose whatever reader or two I might still have left. One encounters such work in “Transition to Advanced Mathematics” courses; even veterans of Calc classes sometimes blench.

The good news at this point is that calcuating inverses with the standard textbook notations is, in principle, a pretty simple matter: given a “formula” for f(x), one puts (typically without any explanation, as if by magic) x = f(y), and then uses ordinary “math 102”-style algebra to solve for y; the result is that y = f^{-1}(x); one has computed the “formula” for the inverse of f (and this formula is in the variable x).

The bad news here is that this is one of those areas where many students literally will not listen to reason: one encounters considerable resistance to attempts to explain why this calculation is appropriate—even more than usual, one will tend here to run up against the “just show me how to do it” wall (“ours is not to reason why, just invert and multiply”, as I read somewhere… “function inverses” is hardly the only situation where this type of resistance becomes an issue…). But not wanting to think about why f(x) = y is equivalent to x = f^{-1}(y) (when f is an invertible function)—by “applying f^{-1} to both sides”— is of course tantamount to rejecting the whole idea of an inverse altogether. Or maybe rejecting algebra itself.

And I’m a long way from knowing what to do about it. But it’s something of an article of faith with me that frequently-encountered student pathologies result from improperly-presented material: there is a right way to do this. Or there will be, once I’ve wrestled the son-of-a-bitch to the ground…

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  1. vlorbik

    it occurs to me only now to consider
    introducing my functions-of-the-hour as
    a_1, m_3, and p_5
    in the… very likely never-to-occur…
    first lecture on all this stuff.

    not… adjusting notation slightly…
    the definitions of the
    m_3:=[x/—}3x]
    type (students would riot)
    but rather the
    a_1(x) = x+1
    m_3(x) = 3x
    p_5(x) = x^5
    variety natural for all
    classroom lectures
    for this material.

    one has implicitly introduced
    three entire *classes* of function here:
    a_k, m_k, p_k
    (where k takes values in…
    well… like the corrupt accountant
    says… what do you *want* it to be?
    … let’s say, the reals. depends what
    class; depends what *day* of the class…
    no, wait! i said *implicitly*!
    what the devil should i be *ex*-
    plicit for then about what set?).

    there’ll be students… and teachers…
    and mathematicians… and math
    “educators”… who *prefer* to call
    multiplication-by-three “m_3″…

    and there’ll be those who prefer to call it m….

    in the sense (say) that, working on their own
    without book or master, they’d just *use*
    the one or the other…

    and this is great. let ’em. but in the *lecture*
    i’m imagining, i’d just throw a_1, m_3, and p_5
    up on the board and keep saying “add one,
    multiply by three, power-by-five”
    (with the remark there that i’ll
    go ahead and make here to the
    effect that i chose p for power
    here rather than e for exponent
    and chose it *advisedly*… my
    natural tendency is to call this
    operation “exponentiate by five”;
    i eschew this vernacular so as
    *not* to confuse it with
    “exponentiate *base* five”.
    here is pedagogical content
    knowledge).

    and then, when i go to
    *calculate* with ’em…
    a\circ m
    and that lot…
    i’ll just say, hey look,
    there they are on the board;
    lets just abbreviate
    a_1, m_3, p_5
    by
    a, m, p
    as if… what is actually true…
    it were the most natural thing
    in the world.




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