Answer
(a) $\frac{x}{4x-1}$, domain: $(-\infty,0)U(0,\frac{1}{4})U(\frac{1}{4},\infty)$
(b) $-x-4$, domain: $(-\infty,-4)U(-4,\infty)$
Work Step by Step
(a) Given $f(x)=\frac{1}{x+4}, x\ne-4$ and $g(x)=-\frac{1}{x}, x\ne0$, we have $(f\circ g)(x)=f(g(x))=\frac{1}{-\frac{1}{x}+4}=\frac{x}{4x-1}$. From $4x-1\ne0$, we have $x\ne\frac{1}{4}$. Combine all the conditions, we can find its domain: $(-\infty,0)U(0,\frac{1}{4})U(\frac{1}{4},\infty)$
(b) $(g\circ f)(x)=g(f(x))=-\frac{1}{\frac{1}{x+4}}=-x-4$. Combine all the conditions, we can find its domain: $(-\infty,-4)U(-4,\infty)$
