Many mathematical texts define a multivariable function $f$ in the following way
$$f := f(x,y)$$
However, if we focus on the fact that a function is really a binary relation on two sets, (say the real numbers), the definition would be as follows
$$ f: \mathbb{R}^2 \rightarrow \mathbb{R}$$
This seems to imply that the domain of the function is a set of ordered pairs of the form $(x,y)$.
The set $\mathrm{graph}(f) \subset \mathbb{R}^2 \times \mathbb{R}$, would then comprise ordered pairs of the form $$\left\{((x_0,y_0),a), ( (x_1,y_1),b),\ldots\right\}$$
In line with this, does it not follow that the correction notation for $f$ should be be
$$f := f( (x,y))$$
Few, if any, texts I have come across use this notation, although it appears the correct way to consider the function as a mapping from one set to another.
Best Answer
Technically speaking, you are correct. If I have a function $f:\mathbb{R}^2\to\mathbb{R}$, and I define $p\in\mathbb{R}^2$ to be the point $p=(1,2)$, then logically, there shouldn't be any difference between writing $f(p)$ or $f((1,2))$; and indeed, there is no logical difference. However, $f(1,2)$ is a convenient shorthand, because the extra pair of parentheses provide no added clarity.
I might also add that, in some places in mathematics, it is common to denote the application of a function without parentheses; for example, if $T:V\to W$ is a linear map from a vector space $V$ to a vector space $W$, you'll often see $Tv$ written for the image of an element $v\in V$ under $T$.
Choosing when to distinguish, or not distinguish, things that have "canonical" identifications, such as ordered pairs $((a,b),c)$ and ordered triples $(a,b,c)$, is an important detail in a lot of mathematics.