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?

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    $\begingroup$ I don't accept the premise :) In other words, I don't agree that "and" usually means "plus". A great deal of the time, for example, it means "and"! And the same for "of". Just as "of" means a thousand different things in English (as anyone learning a foreign language can attest to), it can mean different things in math. $\endgroup$ – Greg Martin Sep 23 '11 at 18:34
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    $\begingroup$ Why should $5(x)$ mean $5$ times $x$? The usual notation would be $5x$, or, if one has stock in a parentheses manufacturer, $(5)(x)$. Admittedly we do use $5(x+y)$. One of the difficult things for students is the large collection of notational conventions that we use unconsciously. One solution, good for computers, bad for people, is absolute consistency. The other solution is the same as with any other natural language: exposure to a large amount of speech/text. $\endgroup$ – André Nicolas Sep 23 '11 at 18:49
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    $\begingroup$ Andre: Perhaps this is why I no longer teach algebra, and am now a computer programmer! Greg: "When I make a word do a lot of work like that," said Humpty Dumpty, "I always pay it extra." $\endgroup$ – cobaltduck Sep 23 '11 at 20:10
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    $\begingroup$ Why not call 5 a function that yields five of its argument? This works in natural language as well. $\endgroup$ – Jon Purdy Sep 23 '11 at 22:09
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    $\begingroup$ I would respond to the student by saying that this kind of verbal-match-up strategy is a terrible way to approach math and science. Now that the student is in 8th or 9th grade, it's time to grow up to a more mature intellectual level. $\endgroup$ – Ben Crowell Dec 31 '12 at 2:17

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.

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    $\begingroup$ I don't think this is specific to English prepositions. People often complain about the arbitrariness of prepositions in languages they're learning until you point out to them the arbitrariness of prepositions in their native language. $\endgroup$ – joriki Sep 24 '11 at 13:37
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    $\begingroup$ @joriki: Fair enough. Coming from Chinese, I find it's trying to map different pronoun systems onto one another that causes the discomfort. Maybe that's true as we think about math in oral language, too. $\endgroup$ – brannerchinese Sep 27 '11 at 11:21

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. :-)

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    $\begingroup$ Also, think "the height of Mary/Jim/Susan/'Person $X$'", or "the density of Air/Water/Diamond/'Substance $X$'", or "the color of Leather/Grass/Oil/'Material $X$'", or "the capital of Germany/France/USA/'Country $X$'", or "the hit-record-count of Presley/McCartney/Yankovic/'Performer $X$'", or ..., or ..., or ... . In the phrasing "Property $F$ of Object $X$", the "of" makes pretty good sense; far more sense than "to" or "applied to" or "at" or "evaluated at", which represent more how mathematicians eventually come to think of functions, but are a bit jargony. $\endgroup$ – Blue Sep 27 '11 at 12:06

The notation $f(x)$ is almost universal but it is by no means the only notation. Witness Mathematica's choice of f[x] also used in M-expressions by McCarthy in the creation of Lisp and the common $\log x$ with no parentheses. See also http://en.wikipedia.org/wiki/Function_application.

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    $\begingroup$ The really neat thing about Mathematica is that functions can be "generic", e.g. (Log[1+#])&, and that a function f can be applied in more than one way: f[x], f @ x, or x // f... $\endgroup$ – J. M. is a poor mathematician Sep 24 '11 at 13:20
  • $\begingroup$ @J.M.: Lisp did this in 1958: en.wikipedia.org/wiki/Lisp_%28programming_language%29 $\endgroup$ – Ben Crowell Dec 31 '12 at 2:15
  • $\begingroup$ @Ben, yes, and that nice bit of LISP was borrowed by Mathematica; the docs for Mathematica say as much about what it owes to LISP. $\endgroup$ – J. M. is a poor mathematician Jan 16 '13 at 17:42

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