Answer
(a) $ x+2$, domain $\{x|x\ge2 \}$.
(b) $ \sqrt {x^2+2}$, domain $\{x|x=all\ real\ numbers \}$.
(c) $ x^4+8x^2+20$, domain $\{x|x=all\ real\ numbers \}$.
(d) $ \sqrt {\sqrt {x-2}-2}$, domain $\{x|x\ge6 \}$.
Work Step by Step
Given $f(x)=x^2+4$ and $g(x)=\sqrt {x-2}$, we have:
(a) $f\circ g=(\sqrt {x-2})^2+4=x+2$, domain $\{x|x\ge2 \}$.
(b) $g\circ f=\sqrt {(x^2+4)-2}=\sqrt {x^2+2}$, domain $\{x|x=all\ real\ numbers \}$.
(c) $f\circ f=(x^2+4)^2+4=x^4+8x^2+20$, domain $\{x|x=all\ real\ numbers \}$.
(d) $g\circ g=\sqrt {\sqrt {x-2}-2}$, domain $\{x|x\ge6 \}$.
