Using the quadratic discriminant

We have found the range of functions by graphical methods so far. For example, the following is a sketch of the graph of $y = f (x) = \frac{2 x}{x^{2} + 1} .$

The curve has a minimum point at $(- 1, - 1)$ and a maximum point $(1, 1) .$ Hence the range of the function is $[- 1, 1] .$

An algebraic method to find the range

Instead of the graphical method above, we can find the range of some functions using the quadratic discriminant.

The idea behind this method is that the range of a function is the set of values of $y$ for which the equation $y = f (x)$ has real roots in $x .$ If we can manipulate $y = f (x)$ into a quadratic equation in $x,$ then we will be able to use the quadratic discriminant to find the range of the function.

For our earlier example, the range of the function corresponds to the horizontal lines that cut the curve $y = f (x)$ at one or two points. This corresponds to the case that our quadratic discriminant $b^{2} - 4 a c \geq 0 .$

Example

$\begin{matrix} y = \frac{2 x}{x^{2} + 1} \\ y x^{2} + y = 2 x \\ y x^{2} - 2 x + y = 0 \end{matrix}$

For the range of $f,$

$\begin{matrix} b^{2} - 4 a c \geq 0 \\ (- 2)^{2} - 4 (y) (y) \geq 0 \\ 4 - 4 y^{2} \geq 0 \\ y^{2} - 1 \leq 0 \\ (y + 1) (y - 1) \leq 0 \\ - 1 \leq y \leq 1 \end{matrix}$

Hence the range of $f$ is $[- 1, 1] .$