Answer
See the proof below.
Work Step by Step
We can rewrite $ f(x)$ as follows
$$ f(x)=x^2+\frac{3}{x}=\frac{x^3+3}{x}$$
which is a quotient of polynomials; that is, $ f(x)$ is rational. Also, for $ g(x)$, we have
$$ g(x)=3x^3-9x+\frac{1}{x^2}=\frac{3x^5-9x^3+1}{x^2}$$
which is a quotient of polynomials; that is, $ g(x)$ is rational.
