What does y=y(x) mean? In many diciplines that utlizes mathematics, we often see the equation
$$y=y(x)$$
where $y$ might be other replaced by whichever letter that makes the most sense in context. My question is what does $y$ mean in this case. I think that $y$ means both a function, since $y(x)$, but also a variable whose value is equall to the output of function $y$. Is that correct?
 A: If you mean you have seen the exact symbols $$y=y(x)$$
rather than $y(x)$ being $x^{2}$ or $x\log x$ or something like that, then this isn't really an equation. It's just the author stressing that the variable $y$ depends on $x$ and on nothing else. For example, if I wanted to describe a conservative potential $\phi$ in 1 dimension, I might write $$\phi=\phi(x)$$
to stress that $\phi$ is a function of $x$ (space) but not $t$ (time)
A: Your perception is perfectly correct. The notation
\begin{equation}
y = y(x)
\end{equation}
is overloaded as $y$ can now be used to refer both to a function $y : D \rightarrow C$ and a variable $y \in C$. This is a dangerous practice which can cause students a world of pain. Consider the following elementary application of the chain rule. Let
\begin{equation}
f : \mathbb{R}^2 \rightarrow \mathbb{R}, \quad x, y : \mathbb{R} \rightarrow \mathbb{R}
\end{equation}
be differentiable, and consider the function $z : \mathbb{R} \rightarrow \mathbb{R}$ given by
\begin{equation}
z(t) = f(x(t),y(t)).
\end{equation}
Then $z$ is a differentiable function and
\begin{equation}
z'(t) = f_x(x(t),y(t))x'(t) + f_y(x(t),y(t))y'(t)
\end{equation}
where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to the free variables $x$ and $y$, whereas $x'$ and $y'$ are the derivatives of the functions $x$ and $y$ with respect to $t$. If at all possible, I prefer not to overload notation.
A: It's just a convention, people decided that this y = f(x) means, the value of y at each value of the function f by changing x, in your case the name of the function is y, but you can call it whatever you want, f(x),y(x),t(x),z(x),cat(x),dog(x)... ecc. In your case you have called your function with the same graphical symbol of the result.
A: If $y=y \, x$, then $y=0$,
$0=0 \, x$, this is the only actual time $y = y \, x$.
