How to answer a student objection to the use of "of" in pronouncing f(x)?

Once upon a time in elementary school, a student learned how to translate certain English words into math. For example, 'and' usually means 'plus' such as "If John has 3 oranges AND 5 apples, how many pieces of fruit does he have?" means to do $3 + 5$. Similarly, 'of' usually means 'multiply' such as "... two groups of 12 things.." or "... a third of 9" indicate $2\times12$ and $1/3 \times 9$ respectively.

Now foward a handful of years and this student finds himself in my algebra course, where I introduce function notation $f(x)$ and state that this is custmarily spoken as "f of x" but caution that even though we write the two symbols side-by-side, that this notation is supposed to suggest action by the function, and not multiplication, unlike $5(x)$ which does mean $5$ times $x$.

The student raises his hand and points out the obvious duplicity of this nefarious two-letter preposition. I mumble something about how mathematics is famous for abuse of the notations it invents, and move on.

How would you have responded to the student?

The notation $f(x)$ denotes the image of $x$ under the function $f$. This is not restricted to abstract symbols like $f$; it's a generalization of expressions like "the square of $x$", "the cube of $x$", "the square root of x", "the cosine of $x$", etc. A function assigns a unique function value to each element of its domain, and this is the function value of that element.

Other options might have been "$f$ to $x$", short for "$f$ applied to $x$", or "$f$ at $x$", short for "$f$ evaluated at $x$", but "of" has the advantage of generalizing the above familiar expressions.

(Also I would have told the student to keep up the critical thinking. :-)

English prepositions are notorious for their seeming arbitrariness and in fact more of them than you would think are variable.

Examples: Are these prepositions different from or to one another? And what do you expect of or from your students when they ask you questions like these? Are you ever angry at or with them? Do you ever think you are going to die of or from embarrassment when they ask you things you can't answer? Have you considered chatting to or with your colleagues about these questions?

As for your actual question, if you have a little spare time, a useful assignment might be to ask students to think about alternatives to "of' in this usage. I would say that in matters of custom, we have to speak in a way that other people can understand, but that I myself would prefer "applied to" or "at" or "imposed on" if I were free to redo things.

Edit, on further reflection (20110927): Perhaps it is the partitive sense of "of" that is operative here — the "of/from" sense. I'm thinking that we can say "log x" but also "the log of x". Why partitive? Because we can make a list of different functions that are applied to x, and then name one of them as though it is being summoned "from" that list. I think something similar is in play in how methods are considered to belong to an instance of an object in OOP.