It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean:
Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also suppose I have a dependence between $x$ and $y$ which is another variable. Say $x = 2y$.
I seems to me I can interpret $f$ and particularly the symbol $f(y)$ in two different ways:
$f$ is strictly a map and it taken whatever variable we give it and maps it accordingly. In this case $f(y) = y$ as $f$ is the identity map so just maps $y$ to itself.
$f$ is a variable dependent on $x$, since I've defined $f(x) = x$, then when I denote $f(y)$ I can interpret it as the variable $f$ but now expressed in terms of $y$ instead of in terms of $x$, so in this case $f(y) = x = 2y$. In this case I viewed $f$ as already being defined in terms of x and the symbol $f(y)$ merely gives me this predefined variable $f$ in terms of $y$.
My question is what is the usual interpretation, and if there is convenient notation and/or common terminology which distinguish the two.