Consider the irreducible rational function in $\mathbb{R}^2$.
$$y=\frac{A(x)}{B(x)}$$
where at least one term is quadratic and the other term has degree either 0, 1 or 2.
The classic way of finding the range of these rational functions is to re-arrange the equation as a quadratic in $x$, and then let the discriminant be greater than or equal to zero. Upon solving this inequality for $y$, one acquires the range of this rational function.
My question is, does this method always work? Is there a counter-example floating out there?