What's the formal definition of saying "$f$ is a function of $x$, $g$ is a function of $y$"?

Question

I know the following definition of function (amongst others).

Let $A,B$ be sets. A set $f\subseteq A\times B$ is said to be a function from $A$ to $B$ if the following two properties hold.

• For every $a\in A$ there exists a $b\in B$ such that $(a,b)\in f$.

• If $(a,b)\in f$ and $(a,c)\in f$, then $b=c$.

In such cases we write $f:A\to B,\, f(a)=b$ .

This definition clearly says nothing about '$f$ being a function of $x$' (or $a$, or whatever you want to call your variable).

So what do people really mean when they say that "$f$ is a function of $x$"?

Answer 1

They denote the argument of the function by any $x \in A$. The name of the variable you pick is not important, it just represents generic input into the function $f$.