Right.

Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices.

It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.

Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.

And so it is.

edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use.

But I think we often thrown them in unintentionally because we're so used to it.

You used too little context when arguing with an audience who isn't used to that style of writing...

Also, there are actual quite reasonable rules to know which 3rd declension nouns are i-stems, so it doesn't seem too right too just say "follow your gut".

Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^

x(3) is a function "x" being applied to the number 3, because "three times x" is 3x.

Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)

I'm sure some would take it that way as "x" isn't commonly used in mathematical notation as a function name, the norm is to use f, g, etc.

However, the notation is relatively clear notwithstanding. It almost has to be x as a function with an input of 3.

I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.

It depends on the context. As an intermediate step I definitely write things like x (3) meaning multiplication, as it can more clearly indicate what's just happened (see my other comment in this thread).

There's other context too, based on what is known. Up to a certain point in first year at my university, most engineering students haven't ever seen functions named x and y, and so they'd mostly interpret x(3) as multiplication. Then we show them parametric curves, and suddenly x(3) looks like the x coordinate of a point on the (x(t), y(t)) curve.

This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f(x)=fx=f×x while f[x] means “apply the function f to the argument x”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.

It's not devoid of ambiguity, because f[A] usually means the image of the set A under f.

Not at all. f[x] is clearly the xth component of the array f :-D

On the blackboard in the mathematics department it’s not.

But I get your point.

I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.

Square brackets can also denote the floor function (especially in old works where '⌊' and '⌋' were typographically unavailable).

An array is merely a function from N to some set.

If you meant "function", would you generally write 𝑓(3) instead of f(3)?

(I have no mathematics background past K-12)