Is $f(x) = \dfrac{3x}{x^{-2}-4}$ a rational function?
On the one hand, the denominator of $f(x)$ is not a polynomial, so $f$ is not a rational function.
On the other hand, we can multiply both the numerator and the denominator of $f(x)$ by $x^2$ to get $g(x) = \dfrac{3x^3}{1-4x^2}$, which is clearly a rational function.