### DM Seeks ODE For ???

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” 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 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 , or (our soon-to-be-introduced) . Because once we start seeing *functions* as objects to be “operated on” (and denote one *operator* on such objects by —call it [for now] “circle”), why then we can start setting up *equations* like 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 (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 (*i.e.*, *f* and *g* are real-valued functions of a real variable [see?]). We then define **the composite of f with g** by

(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:

(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 … 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 and , the reader may be in a position to compute (formulas for), say, and , as well as to observe that in general —function compositon is not commutative. Ambitious readers can consider “inverse” functions like

and

,

going on to show that —the inverse of the composite is the composite of the inverses *in the opposite order* (the “shoes and socks” theorem).

There’ll be much more of this anon. Or there would if I had a better attention span.

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