Following a thorough discussion of such examples, let us define “expressions” — (words in +,- ,.,/, radical,x,y,z, etc), and then define equations as relationships between expressions. And the let us LIE! a function is an equation in which one side is y, and the other side is an expression in x. TELL STUDENTS THAT IT IS A LIE, but explain to them that
it is an operational definition.

Then after weeks and weeks of algebraic examples of rational, trig, exponential, logarithmic examples, discuss functions as a relationship between two sets so that for each x here is a unique y.

Abstraction can follow easily from example.

I had forgotten the distinction between relation and function. Thanks for pointing that out. Functions are a sub-set of relations which passes the vertical test. Both can be described as abstractions of mathematical expressions (though relations appear to have even looser constraints than that, i.e. just a list of ordered pairs).

I’d love to be able to say that relations report data, and functions extrapolate further data based on the relation, but that just doesn’t seem to cut it, either. Can you recommend any further reading for me?

Walt:
Primarily, I didn’t grasp the problem solved or question answered by functions. I didn’t have a peg on which to hang the hat. The notation, by itself, was not the problem. It was translating it back into English, actually my inability to do so, that kept me back.

The introduction of functions appears to be a point where mathematical syntax diverges from lingual expression enough to make it a hang up for literary types like me.

The idea of `function’ is nothing trivial. I am not even sure why it is introduced so early in such abstract terms. Before the 20th century there was no abstract notion of mappings, an d words like `function’ (or its translation in French or German) generally meant something like a locally defined analytic function, that is something given locally by a power series. Riemann surfaces were invented as a way of coping with multivalued functions (it’s normal to teach pre-calculus students something going under some silly name such as the `vertical line test’ that says a function cannot be multivalued - but mathematicians use multivalued functions all the time - this is essentially what a closed differential one-form that is not exact is) and it was only after an adequate global geometric language was found that the modern abstract notion of function was formalized in the way we know it now. My point is that the way we talk about functions, even to pre-calculus students, is informed by the categorical/functorial point of view, which is rather sophisticated (at least we don’t say `epimorphism’ to pre-calc students). What’s insane is that we are teaching folks what is a function who have not yet first mastered manipulations with polynomials and trigonometric functions (by the way, to treat the latter in any kind of adequate way without using calculus, it is necessary to present quite a lot of elementary geometry which the typical pre-calc student has not seen at all because of the atrophied presence of plane geometry in the standard curriculum - the power series definition is unmotivated and computationally useless (at least at first) - and differential equations aren’t available - so one has to speak directly in terms of angles - this suffices to define the trigonometric functions - but how to compute them? the work is quite serious, even if to us now it seems easy). Moreover, the abstract notion of `function’ is wholly unmotivated. Why should a student care?