DM Seeks ODE For ???

February 17, 2009 in Math 148

In the AM classes, I began with bigpicture stuff (for reasons obscure to me): function composition considered as an operation. Specifically, as on operation on functions.

The point here is to stress the analogy with the more familiar situation where operations like addition, subtraction, multiplication, and division are applied to (pairs of) numbers to produce “new” numbers, the operation—composition of functions—we are about to consider will be applied to (pairs of) functions to produce new functions.

Just as the “code” $a \times b$ denotes a number when a and b are numbers, so too, whenever instead of numbers, we’re considering functions, we can (and will) denote by $f\circ g$ a new function—which is still to be defined here.

And why have I been delaying this crucial definition? Because I’m trying, indeed I can only hope trying not-too-desperately, to call your attention specifically to the perfect parallel in the way the notation is layed out.

Let me now go on, after digressing to remark that by now we’re pretty far away from anything rightly called “math 148 lecture notes”, to name this “layout”. Let’s call it object-infix-object. By “object”, one more-or-less obviously means number-or-function and by “infix” an operation symbol like $+\,,-\,,\times\,,\div$ , or (our soon-to-be-introduced) $\circ$ . Because once we start seeing functions as objects to be “operated on” (and denote one operator on such objects by $\circ$ —call it [for now] “circle”), why then we can start setting up equations like $h = f\circ g$ and start right in solving ’em: we are in the presence of an Algebra of Functions.

And only by getting away from numbers can we clearly see what’s been happening all along, from arithmetic on up. Quite often, problem-solving techniques developed for one application (equations about numbers, say) will turn out to be useful in other applications (equations about functions, for example). Here is power: the proper study of mathematics is not counting or measuring or even necessarily calculating but reasoning itself. Sets-With-Operations turn out to be exactly the right framework for an enormous variety of problems: a framework whose power and flexibility are awe-inspiring (pretty much universally: this is “math-phobia” laid bare).

Returning (slowly) to actual course-like material, let me mention here that I do sometimes let slip there a reference to the “Real (or Complex) Field” (instead of the “set of Real (or Complex) Numbers”), for example; I’m even more prone to write down things like $f, g \in {\Bbb R}[x]$ (typically with an immediate gloss [“f & g polys w/ Real coeffs” or somesuch])—having the right symbols is even more important than having the right words (and everybody knows I’m a fanatic for having the right words).

Suppose $f, g : {\Bbb R} \rightarrow {\Bbb R}$ (i.e., f and g are real-valued functions of a real variable [see?]). We then define the composite of f with g by
$f \circ g (x) = f(g(x))$
(whenever the right-hand-side is, itself, defined—as it is not when, for example, g is the “constant at zero” function and f the “reciprocal” function).

Let’s go ahead and fix a few functions for purposes of illustration:
$a(x) = x+ 1$
$m(x) = 3x$
$p(x) = x^5$
(the “add-one” function, the “multiply-by-three” function, and the “raise-to-the-power-five” function).

Let me say here that a is not “x + 1” [a(x) is; a itself is a function not a number]. Maybe we can throw some light on this by writing $a = [x \mapsto x+1]$ … note that this is entirely in the spirit of my stated program of creating an “algebra” for dealing with equations about functions.

It may be nothing more than a matter of taste, but I find it much more satisfying to define a function called a with an equation beginning “a=”, rather than (the much more common) “a(x)=“—as if the name of the variable had anything to do with the function itself. I’ve ranted about this before.

All of which grades no papers and that guitar’s not gonna start playing itself. So I’ll wrap up.

It’s not a coincidence that all three of the examples I chose have as their domain and range the full set of Reals (but neither is it a necessity; I’ve chosen simple examples to begin our investigation); unusually alert readers (mostly having had a course like 148 already) may have noticed that each is also (what we well later call) a one-to-one function.

Given the models $m\circ a (x) = 3x+3$ and $a\circ m (x) = 3x + 1$ , the reader may be in a position to compute (formulas for), say, $m\circ p$ and $p \circ m$ , as well as to observe that in general $f\circ g \not= g\circ f$ —function compositon is not commutative. Ambitious readers can consider “inverse” functions like
$a^{-1}(x) = x -1$ and
$m^{-1}(x) = {x\over 3}$ ,
going on to show that $(a\circ m)^{-1}= m^{-1}\circ a^{-1}$ —the inverse of the composite is the composite of the inverses in the opposite order (the “shoes and socks” theorem).