Answer
a. $(f\circ g)(x)=\sqrt {-2x-1},$
b. $(g\circ f)(x)=1-2\sqrt {x-2},$
c. $(f\circ f)(x)=\sqrt{\sqrt{x-2}-2}$
d. $(g\circ g)(x)=4x-1$
Work Step by Step
$f(x)=\sqrt {x-2},$
$g(x)=1-2x,$
a. $(f\circ g)(x)=\sqrt {1-2x-2}=\sqrt {-2x-1},$
Domain of $(f\circ g)(x)$ is, $x\leq-\frac{1}{2},$
b. $(g\circ f)(x)=1-2(\sqrt {x-2})=1-2\sqrt {x-2},$
Domain of $g\circ f(x)$ is, $x\geq2,$
c. $(f\circ f)(x)=\sqrt{\sqrt{x-2}-2}$
Domain of $(f\circ f)(x)$ is, $x\geq6,$
d. $(g\circ g)(x)=1-2(1-2x)=1-2+4x=4x-1,$
Domain of $g\circ g(x)$ is, $x\in \mathbb{R}$
