What’s the difference between “the value x” and “the value of x”? What’s the difference between “the value $x$” and “the value of $x$”?
I’m from Poland. I read scientific articles and can’t figure out the difference. Some sample sentences: Thus, a function $f$ should be distinguished from its value $f(x_0)$ at the value $x_0$ in its domain. (…) since $f(x)$ and $x_2$ should both be understood as the value of $f$ at $x$. (…) valid for all real values of $x$ ". (Source: https://en.wikipedia.org/wiki/Function_(mathematics))
Thanks in advance,
Chris
 A: Short answer: I don't think you should worry, as long as you understand this sentence, which says that a function is the whole rule, not a result of applying a rule.

Thus, a function $f$ should be distinguished from its value $f(x_0)$
  at the value $x_0$ in its domain.

tl;dr
In the first quoted sentence each "value" is a named object, $x_0$ in the domain and $f(x_0)$ in the codomain.
The  problem with "values" comes up because functions are often described using a formula with a "variable", usually $x$. The point of the discussion is to make clear that when the function is defined that way, as in "$f(x) = x^2$" , there is really "no $x$" in the definition, 

since $f(x)$ and $x^2$ should both be understood as the value of $f$
  at $x$.

Here you say "value of $f$" because "value $f$" makes no sense. The modifier "value" belongs before a number.
Then this last one is really tricky. Here you say "value of $x$" because you are thinking of $x$ not as a number but as the identity function on the domain. 

valid for all real values of $x$

