Answer
$\{x|x\ge0\ and\ x\ne4 \}$.
Work Step by Step
Step 1. Domain of $g(x)$: $\{x|x\ge0 \}$.
Step 2. For $(f\circ g)(x)$ we need: $g(x)-2\ne0 \Longrightarrow \sqrt x-2\ne0 \Longrightarrow x\ne4$.
Step 3. Thus Domain of $(f\circ g)(x)$: $\{x|x\ge0\ and\ x\ne4 \}$.
