Answer
$(-\infty,-\sqrt 2]U[\sqrt 2,\infty)$
Work Step by Step
Step 1. Given $f(x)=\sqrt {x-2}$ and $g(x)=x^2$, we can find their domains as $[2,\infty)$ and $(-\infty,\infty)$
Step 2. $(f\circ g)(x)=\sqrt {x^2-2}$ which requires $x^2-2\ge0$ or $x^2\ge2$ thus $x\in(-\infty,-\sqrt 2]U[\sqrt 2,\infty)$
Step 3. Combine all the conditions, we have the domain for $(f\circ g)(x)$ as $(-\infty,-\sqrt 2]U[\sqrt 2,\infty)$
