I prefix this is a question about notation, and in particular regarding parentheses.
Let us say we have an equation like this:
$$z=f(x_0,y_0 )+ \frac{∂f}{∂x} (x_0,y_0 )(x-x_0 )+\frac{∂f}{∂y} (x_0,y_0 )(y-y_0 )$$
$(x_0,y_0 )$ indicates that the partial derivatives are evaluated at the point $(x_0,y_0 )$. Although in this case there's a comma that clearly higlights a coordinate pair, in some other cases it may happen to have functions to be evaluated at a single point, such as:
$$z=f(x_0)+ f'(x_0)(x-x_0 )$$
Although it's quite obvious that $f$ and $f'$ should be evaluated at $(x_0)$, I was wondering if there exists or not a way to distinguish the parentheses that refer to a point from the others.
Well, you can insist if $f: \mathbb R^2 \to X$ then as the elements of $\mathbb R^2$ are ordered pairs of the form $(x_0,y_0)$ then one should write $f(\color{\red}(x_0,y_0\color{\red}))$ and not $f(x_0,y_0)$. But one can counter argue that parenthesis are always just a convenience for containment. That an ordered pair of $x_0$ and $y_0$ is just the pair $x_0,y_0$ and the parenthesis only exist to keep your groceries from spilling to the floor. And likewise $f(x_0) = m$ simply means $f$ maps $x \mapsto m$ where as $f(x_0,y_0)$ simply means $f: $ the pair $x_0,y_0 \mapsto m$.
Anyway, this is usually considered too convenient a shortcut to get pedantic over. In any event it is assumed to be clear that if the input of a function is $\underbrace{w_1, w_2, ......,w_n}$ that the domain of the function is $\mathbb R^n$.
A more pertainent issue, to my mind, is is it to be assumed that $f:\mathbb R^2 \to \mathbb R^{\color{red}1}$ and that $z\in \mathbb R$?