Show that a rational function of x (quotient of two quadratics or similar) takes all real values, or find its range, by setting y equal to the expression and analyzing when the resulting quadratic in x has real solutions.
If $c$ is a real number with $0 < c < 1$, then show that the values taken by the function $y = \frac{x^2 + 2x + c}{x^2 + 4x + 3c}$, as $x$ varies over real numbers, range over all real numbers.