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A few times, I wanted to highlight that a variable in the right-hand side of an equation depended on other variable. I can't recall a good example right now, but consider this one:

$\tau = F d(\vec{r})\cos(\theta)$.

Probably, the person who hypotetically wrote probably wanted to mean that $d$ depends on $\vec{r}$. In my experience, many students will think that $d(\vec{r})$ means $d \times \vec{r}$.

That problem would not exist if there were specific parentheses for function arguments.

Leonardo Castro
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    I think this is a perfectly reasonable question that's not opinion-based at all: are there alternatives to the very common function notation? – pjs36 Feb 05 '16 at 13:22
  • Well, I have some reasons to think that my question is valid here. It's a question about notation, a subject that has its own keyword here. It's not opinion-based because it stems from an observed fact: students sometimes take argument function parentheses as a multiplicative factor. I've witnessed that myself. – Leonardo Castro Feb 05 '16 at 13:35
  • So surprised to find almost no discussion of this topic. It seems completely weird that we commonly use this notation with so imprecise definition. – FlorianH Jul 08 '21 at 20:32

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Yes, there is at least one. Murray Bourne, owner of the site Interactive Mathematics, proposes writing boxed arguments:

I wish to propose an alternative notation for concepts where you cannot expand in the way you do with simple algebra. It might look something like this:

$\sin\boxed{x + y}$

$\log\boxed{x + y}$

$f\boxed{x}$

This would send a much clearer message to students that the particular function or operation does not work in the same way as simple algebra works.

[...]

So a more computer friendly option would be to (exclusively) use [ ] - square brackets - for such concepts, like this:

$\sin[x + y]$

$\log[x + y]$

$f[x]$

http://www.intmath.com/blog/learn-math/towards-more-meaningful-math-notation-661

So, you (I?) weren't the first one to think about it. Those proposals don't seem to be very popular, probably because everybody is very used to $f(x)$, which has a long tradition in Mathematics. Surely, expression like

$f(x+1)(x+1)$

might be confusing, but that problem could be eased by having different space lengths for arguments and multiplicative factors, say

$f(x+1)\, \, (x+1)$,

putting factors always before functions,

$(x+1)f(x+1)$,

which, on the other hand, could make long calculations involving multiple factors less clear if the positions were always rearranged to follow that rule.

One could also always write explicit multiplication symbols:

$f(x+1)\cdot(x+1)$

or

$f(x+1)\times(x+1)$.

Having a consistent notation is also helpful: if an $f$ or a $g$ is also followed by a list of arguments between parentheses, people are less likely to interpret that list as a multiplicative factor.

In your example, you could also write

$\tau = F d \cos{\theta}$

where $d := d(\vec{r})$.

If I recall correctly, there are also authors that use always a special font for the letters representing functions, and others who use small size, vertically centered arguments between parentheses, like this:

$f\vcenter{\scriptstyle{(a)}}$,

so that distinction from multiplicative factors is more evident:

$f \vcenter{\scriptstyle{(x+1)}}(x+1)$.

Someone defined a mathmiddlescript macro for LaTeX that could be used for that purpose:

https://tex.stackexchange.com/questions/250961/subscript-superscript-middlescript

Combining styles, we may have interesting options:

$f\vcenter{\scriptstyle{[x+1]}}(x+1)$

$f\,\vcenter{\scriptstyle{[x,\,y\,|\,a,\,b]}}$

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Leonardo Castro
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