Notation of author-defined functions. I have seen functions defined various ways, and I'm wondering which form I should use for some functions. I am hoping to learn what it might be read as, interpreted, etc.
The most common definition of a function is by far $f(x_0,\ldots,x_n)$, for example, $f(a,b,c)=\frac{a^b-c^a}{abc}$. I assume that this is read as "(the) $f$ of $a,b, $ and $c$".
I have seen another definition as $R_z$, for example, $C_n$ is the cyclic graph with $n$ vertices. I have interpreted this to refer to some special element of a larger set, as opposed to $f(x)$, which to me connotes a routine of sorts.
A symbol is often used to denote some function, for example, $\bigcup_{\{x_i\}_{i\in I }}$, $\mathbb{Z}_n$, and $\sum_{i=0}^{12}i^2-i$. I'm not entirely sure when to use this.
The last definition I have seen is $f(x;y)$. I have seen this only in some probability functions, and, with my limited exposure, I interpreted it to mean "$f$ of $x$ with respect to $y$" or something similar.
My question is this: When should I use each notation? Are there any other ways of denoting a function? I wish to define a large number of functions, and I wonder which cases would require each notation.
 A: The notation $f(x)$ is the clearest and probably the easiest way of denoting functions. Things like $f(x;y)$ are sometimes used to denote a function of $x$ with a "parameter" $y$ considered fixed - we make an implicit mental distinction between the "input" $x$ and the parameter e.g. Suppose you have a function $f(x;\epsilon)=x^2+\epsilon$ with a parameter $\epsilon$; this can be thought of as a whole collection of functions of $x$ which vary with $\epsilon$. And this might be a useful idea, because you see by making $\epsilon$ slightly less than zero you obtain two distinct real roots, when it's zero you have a single root, and for positive $\epsilon$ you have no real roots - so the functions $f(x;\epsilon)$ have very different overall properties depending on the value of the parameter. 
Often, if your domain of the function is $\mathbb{N}$ it is commonplace to denote $f(n)$ by $A_n$ where $A_n$ takes values in whatever your codomain is. 
So overall it's a conceptual distinction as to how the values of the function change depending on changes in each of the inputs. Most often, though, you will probably end up using some variant on $f(x)$, incorporating various parameters as needed. 
