Determine if a function is well defined or not

Question

I am doing some mathematics, and I am currently stuck on something. I do not understand this part at all, how can two slightly, minor different functions, but near identical—evaluate so differently.

Exercises:

Do not understand part:

$$(a)\enspace f:R\to R, f(x)=\frac{x}{3},\enspace \it{Well\enspace Defined} $$ $$(b)\enspace g:J\to J, g(x)=\frac{x}{3},\enspace \it{Not\enspace well\enspace defined} $$

Why is $(a)$ well defined, whereas, $(b)$ is not?

Interesting explanation found here, however, I cannot get any smarter.

Reference:

• Discrete Mathematics for Computing, 3rd Edition by Peter Grossman

EDIT 8/1/2023 4:50 PM Clarity. $R$ and $J$ are sets.

EDIT: 8/1/2023 5:25 PM From the book:

• $N$ is the set of natural numbers (or positive integers): $\{1,2,3,4,...\}$.

• $J$ is the set of integers: $\{...,-3,-2,-1,0,1,2,3,...\}$.

• $Q$ is the set of rational numbers: $\{x:x=m/n\enspace for\enspace some\enspace integers\enspace m\enspace and\enspace n\}$.

• $R$ is the set of real numbers.

Answer 1

$\bf{J}$ is Grossman's notation for the set of positive and negative integers (see beginning of the chapter on sets). So obviously $g$ isn't well-defined over that set!