Answer
(a) $ x$, domain $\{x|x\ge1 \}$.
(b) $ |x|$, domain $\{x|x=all\ real\ numbers \}$.
(c) $ x^4+2x^2+2$, domain $\{x|x=all\ real\ numbers \}$.
(d) $ \sqrt {\sqrt {x-1}-1}$, domain $\{x|x\ge2 \}$.
Work Step by Step
Given $f(x)=x^2+1$ and $g(x)=\sqrt {x-1}$, we have:
(a) $f\circ g=(\sqrt {x-1})^2+1=x$, domain $\{x|x\ge1 \}$.
(b) $g\circ f=\sqrt {(x^2+1)-1}=\sqrt {x^2}=|x|$, domain $\{x|x=all\ real\ numbers \}$.
(c) $f\circ f=(x^2+1)^2+1=x^4+2x^2+2$, domain $\{x|x=all\ real\ numbers \}$.
(d) $g\circ g=\sqrt {\sqrt {x-1}-1}$, domain $\{x|x\ge2 \}$.
