Answer
(a) $\sqrt {\frac{1}{x+5}}$, domain: $(-5,\infty)$
(b) $\frac{1}{\sqrt x+5}$, domain: $[0,\infty)$
Work Step by Step
(a) Given $f(x)=\sqrt {x}, x\ge0$ and $g(x)=\frac{1}{x+5}, x\ne-5$, we have $(f\circ g)(x)=f(g(x))=\sqrt {\frac{1}{x+5}}$. From $x+5\gt0$, we have $x\gt-5$. Combine all the conditions, we can find its domain: $(-5,\infty)$
(b) $(g\circ f)(x)=g(f(x))=\frac{1}{\sqrt x+5}$. Combine all the conditions, we can find its domain: $[0,\infty)$