Example 3 Find the range of the function $y=\frac{1}{2 x^{2}+x-1}$.
Solve the functional transformation to
$$
2 y x^{2}+y x-(y+1)=0
$$

When $y=0$, equation (3) is $1=0$, which is obviously not true, $\therefore y=0$ is not in the range of the function;

When $y \neq 0$, $\because$ the equation has real roots, $\therefore \Delta=y^{2}+$ $8 y(y+1) \geqslant 0$, solving this yields $y>0$ or $y \leqslant-\frac{8}{9}$,