So if a function is a process that associates each $x\in X$, known as the domain of the function, a unique element $y\in Y$ known as the the co-domain of the function.
So to outline a few things I think are correct (Please let me know if I have a wrong understanding):
A function is usually denoted by a single letter like $f$, $g$ or $h$.
$f(x)$ is the value of the function $f$ at $x$. You can say $f$ is a function of the variable $x$.
It seems as though, from the comments, that $f$ can be defined by saying what it does to a general element of its domain by stating an identity and then a domain for which it is true. for example:
$f$ is defined by $f(x)=x+1$ for $x>0$
or, $g(x)= x + 1$ for all $x>2$ would define $g$
I think what I have done is write $2$ identities which means that both sides of the relation define the same function of $x$ on the domain stated.
Question 1: So $f$ is the function, $f(x)$ is the value of the function at $x$ and $x+1$ is the value of $f$ at $x$. Is it right that $f(x)$ tells you that $f$ is a function of $x $ and $x+1$ is a function of $x$. I find this hard to understand as I have just said that $x + 1$ is the value of $f$ at $x$.