Answer
(a) $ \frac{4}{x+4}$, domain $\{x|x\ne-4,0 \}$.
(b) $ \frac{-4x+4}{x}$, domain $\{x|x\ne0,1 \}$.
(c) $ x$, domain $\{x|x\ne1 \}$.
(d) $ x$, domain $\{x|x\ne0 \}$.
Work Step by Step
Given $f(x)=\frac{x}{x-1}$ and $g(x)=-\frac{4}{x}$, we have:
(a) $f\circ g=\frac{-\frac{4}{x}}{-\frac{4}{x}-1}=\frac{4}{x+4}$, domain $\{x|x\ne-4,0 \}$.
(b) $g\circ f=-\frac{4}{\frac{x}{x-1}}=\frac{-4x+4}{x}$, domain $\{x|x\ne0,1 \}$.
(c) $f\circ f=\frac{\frac{x}{x-1}}{\frac{x}{x-1}-1}=x$, domain $\{x|x\ne1 \}$.
(d) $g\circ g=-\frac{4}{-\frac{4}{x}}=x$, domain $\{x|x\ne0 \}$.
