Yes, that is perfectly valid. It is a little "unusual" in that, while the range of f certainly is a subset of R, writing f:N->N (or even f:N->E, the set of even integers) would be more accurate.

I would consider this incorrect since "x" is not specified on the left. Perhaps f: x|-> 2x would work.Or, must one write: f:N->R, f|->2x

I wouldn't. I would interpret "|->" as meaning "goes to" or "changes to" and f(x) is already 2x.Would one ever write: f:N->R, f(x)|->2x

Again, I would prefer "x|-> 2x".

Strictly speaking that is false but it is a commonly used "abuse of notation".True or False

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

With the unstated understanding that the domain of f is the largest set for which that formula makes sense, yes. (Here that would normally be either the real numbers or complex numbers depending upon the "domain of discourse".) Strictly speaking a function is defined by both a formula2. It is proper to say that the function “f” is defined by f(x) =2x.anda domain.

No, that wouldn't make any sense to me.3. It is improper to say that the function “f” is defined by f(x).

With the understanding that "f|->2x" (as above I would prefer f:x|-> 2x) implies that f(x)= 2x and the same understanding about the domain as above, yes, that is acceptable.4. It is proper to say that the function “f” is defined by f|-> 2x.

No, it is not. The "image" of a function is the5. It is proper to refer to “f(x)” as the image of “f”.setof all values of f. I would accept { f(x)| x in whatever domain f has} as the image. Personally, I have always preferred f(A)= {f(x)| x A} as the "image of f on A" with the set A specified.

Same answer as above with, I presume, f(x)= 2x.6. It is proper to refer to “2x” as the image of “f”.

I have never seen "characteristic function" used in that way! Why talk about the characteristic function of a7. It is proper to refer to f(x) = 2x as the “characteristic function" of “f”.function?

No!7a. It is proper to refer to f(x) = 2x as the “characteristic equation" of “f”.

No, that's simply an improper use of the phrase "characteristic function" (in English. Are you translating this from another language? Perhaps "characteristic" is not the right word.)8. It is proper to refer to f|-> 2x as the “characteristic function of “f”.

No. Same as above with the added objection that "f|-> 2x" is not an "equation"!8a. It is proper to refer to f|-> 2x as the “characteristic equation of “f”.

First, I would not say "domain set", I would just say "domain". The "domain" of a function is already a set. Second, whether it is proper or not would depend upon whether or not the set of all natural numbers9. It is proper to say that N is the domain set of “f”.wasthe domain of f! Are we still talking about f:N->R, f(x)= 2x?

Different texts may say different things about this. Some will say that the range of f:N-> R, f(x)= 2x has range R, others would say the range is the set of all positive, even integers. Personally, I would say that R is the range and E, the set of positive even integers, is the "image".10. It is improper to say that R is the range set of “f”.

No. The codomain is the same as the image.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.