I would write $\mathrm{dom}\,(f)=\mathbb{R}\setminus\{3\}$ for the domain. For the image notations vary; I would refer to the range as the image and write $\mathrm{im}\,(f)=\mathbb{R}\setminus\{0\}.$ Alternatively, many people prefer notation such as $f(\mathbb{R}\setminus\{3\})$ or $R(f)$ for the range.

Note that the way you have written the range is incorrect only because you haven't written the intervals correctly. What I presume you meant is $$(-\infty,0)\cup(0,\infty),$$ which is correct.