Answer
(a) $ \sqrt {-2x-1}$, domain $\{x|x\le-\frac{1}{2} \}$.
(b) $ 1-2\sqrt {x-2}$, domain $\{x|x\ge2 \}$.
(c) $ \sqrt {\sqrt {x-2}-2}$, domain $\{x|x\ge6 \}$.
(d) $ 4x-1$, domain $\{x|x=all\ real\ numbers \}$.
Work Step by Step
Given $f(x)=\sqrt {x-2}$ and $g(x)=1-2x$, we have:
(a) $f\circ g=\sqrt {(1-2x)-2}=\sqrt {-2x-1}$, domain $\{x|x\le-\frac{1}{2} \}$.
(b) $g\circ f=1-2\sqrt {x-2}$, domain $\{x|x\ge2 \}$.
(c) $f\circ f=\sqrt {\sqrt {x-2}-2}$, domain $\{x|x\ge6 \}$.
(d) $g\circ g=1-2(1-2x)=4x-1$, domain $\{x|x=all\ real\ numbers \}$.
