Answer
$\{x|x\ne-\frac{2}{3},0\}$.
Work Step by Step
Step 1. Domain of $g(x)$: $\{x|x\ne0 \}$.
Step 2. For $(f\circ g)(x)$: $g(x)+3\ne0 \Longrightarrow \frac{2}{x}+3\ne0 \Longrightarrow x\ne-\frac{2}{3}$.
Step 3. Thus Domain of $(f\circ g)(x)$: $\{x|x\ne-\frac{2}{3},0\}$.
