functionsnotation
I have this equation
$$f(x)=\frac{2}{x-3}$$
My thoughts is that I would state the domain like this
$D:x\in\mathbb{R},\ne3$
and the range like this
$R:(0,-\infty) \cup (\infty,0)$
I think I express the domain wrong, but I range I know I got right.
How would I express the range, if I wrote the range wrong?
Notations vary. You should pick something straightforward, and state it clearly up front. One common choice would be $$\operatorname{dom} f\\ \operatorname{ran} f$$
another reasonable choice might be $$\mathscr{D}(f)\\\mathscr{R}(f)$$
Principia Mathematica used:
$\lbrace x\in \mathbb{R}, x\neq [-3,-1,2]\rbrace$ doesn't really make sense because $x$ is a real number, so it can't be equal to $[-3, -1, 2]$ regardless of what that notation means. If you want to use set notation like that, an alternative would be $$\lbrace x\in \mathbb{R} \mid x\not\in \lbrace -3, -1, 2 \rbrace\rbrace,$$ which is exactly what you were trying to express, but with proper set notation. Another way you could write it is $$x\in \mathbb{R}\setminus \lbrace -3, -1, 2\rbrace,$$
which means all of $\mathbb{R}$ without the set $\lbrace -3, -1, 2\rbrace$.
Best Answer
You can write the domain as $x\in\mathbb{R}\backslash\{3\}$, which is the more "normal" convention and the range $x\in\mathbb{R}\backslash\{0\}$ or $(-\infty , \infty)\backslash\{0\}$ or even $(-\infty,0)\cup(0,\infty)$.