TITLE: Abuse of notation in declaring a variable is a function of another?
QUESTION [6 upvotes]: The standard way to write $ \text{ y is a function of x} $ is
$ y = f(x) $
This is taken to mean that $y$ is the value of function $f$ evaluated at $x$. For simplicity let's take $f$ to be some mapping, $ f:\Bbb R\to\Bbb R$.
I cannot understand if mathematics authors are justified in using the notation
$y = y(x)$
to declare that $y$ is a function of $x$. The reason is a type mismatch, it cannot be be possible for $y$ to be a binary relation, as well as some element in the codomain of the binary relation.
Is the notation above commonly accepted? I have seen it in a few published papers, and am not sure whether it is an abuse or has some sound mathematical reasoning.

REPLY [3 votes]: This bothers me too and I never do it in my private life. However, when teaching calculus it's unavoidable. To keep my inner-stickler happy, I secretly think to myself that $x=\DeclareMathOperator{id}{id}\id_X$ where $X$ is the domain of $y$. Thus we have
$$
y=y\circ\id_X=y\circ x
$$
Defining $y(x)$ as $y\circ x$ then justifies the notation $y=y(x)$ and I don't have to think about elements.