Answer
See below.
Work Step by Step
$h(f(x))=h(3x^{-1})=h(\frac{3}{x})=\frac{\frac{3}{x}+4}{3}=\frac{\frac{3+4x}{x}}{3}=\frac{3+4x}{3x}$
The denominator of this function's first term cannot be $0$, thus $x\ne0$. Thus, the domain is all real numbers apart from $x=0$.
