Perform polynomial long division of the numerator by the denominator.

We want to divide $2x^3 - x$ by $x^2 - 1$.

$$\begin{array}{c|cc cc} \multicolumn{2}{r}{2x} \\ \cline{2-5} x^2 - 1 & 2x^3 & +0x^2 & -x \\ \multicolumn{2}{r}{-(2x^3} & & -2x) \\ \cline{2-4} \multicolumn{2}{r}{0} & & x \\ \end{array}$$

The quotient is $2x$ and the remainder is $x$.

This means $f(x) = \text{quotient} + \frac{\text{remainder}}{\text{divisor}}$.

$$f(x) = 2x + \frac{x}{x^2 - 1}$$

Comparing this to the form $g(x) + h(x)$, we have:
$g(x) = 2x$
$h(x) = \frac{x}{x^2 - 1}$

The function $f(x)$ expressed in the form $g(x) + h(x)$ is:

$$2x + \frac{x}{x^2 - 1}$$