Now parentheses "()" is used for both function, e.g. $f(x)$, and for order of operations, e.g. $(3+5)*2$.
Ever in math history be suggested to differentiate? for example square bracket "[]" can be dedicated to order of operation, and "()" only for function?
this is natural as in primary schools we were taught to use (), [], then {} for order of operation, limiting to [] won't cause any confusion.
just a wild thought...
Elegance of notation, even at the expense of some easy-to-resolve ambiguity is important. To see why everybody is OK with using $f(x)$ and also $(c+d)(a+b)$ even though $c(a+b)$ becomes ambiguous, consider that a mathematician steeped in analysis would write something like $$ f : \Bbb{R} \rightarrow \Bbb{R} | \forall x \in \Bbb{R} f(x) = x^2 $$ where some more sloppy people might just write $$f(x) = x^2$$
This might illustrate why too much worry about precision in notation is good for some purposes but not for all purposes.
At any rate, certainly before the time of Fermat, the notation $f(x)$ was not in use; and it was surely in use by the time of Gauss. Not being a historian, I don't knwo where it originated.