Syntax clarification.

I don’t mean to be splitting hairs but these things I am wondering about.

Is this formally acceptable as a definition of “f”, f:N->R, f(x)=2x

Or, must one write: f:N->R, f|->2x

Would one ever write: f:N->R, f(x)|->2x

True or False

1. It is improper to say "the function f(x)”.

2. It is proper to say that the function “f” is defined by f(x) =2x.

3. It is improper to say that the function “f” is defined by f(x).

4. It is proper to say that the function “f” is defined by f|-> 2x.

5. It is proper to refer to “f(x)” as the image of “f”.

6. It is proper to refer to “2x” as the image of “f”.

7. It is proper to refer to f(x) = 2x as the “characteristic function" of “f”.

7a. It is proper to refer to f(x) = 2x as the “characteristic equation" of “f”.

8. It is proper to refer to f|-> 2x as the “characteristic function of “f”.

8a. It is proper to refer to f|-> 2x as the “characteristic equation of “f”.

9. It is proper to say that N is the domain set of “f”.

10. It is improper to say that R is the range set of “f”.

11. It is proper to say that R is the codomain set of “f”.

My guess is that all of the foregoing are true, but I wouldn't bet even a small pig let alone the farm.