# How do you find domain and range for f(x)=(x+6)/(x^2+5) ?

May 13, 2018

The domain is $x \in \mathbb{R}$. The range is $y \in \left[- 0.04 , 1.24\right]$

#### Explanation:

The denominator is always $> 0$ whatever $x \in \mathbb{R}$

The domain is $x \in \mathbb{R}$

To find the domain, proceed as follows

Let $y = \frac{x + 6}{{x}^{2} + 5}$

Rearranging,

$y \left({x}^{2} + 5\right) = x + 6$

$y {x}^{2} - x + 5 y - 6 = 0$

This is a quadratic equation in $x$ and in order for this equation to have solutions, the discriminant $\ge 0$

$\Delta = {\left(- 1\right)}^{2} - 4 \cdot y \left(5 y - 6\right) = 1 - 20 {y}^{2} + 24 y$

$- 20 {y}^{2} + 24 y + 1 \ge 0$

The solutions to this inequality is

$y \in \left[\frac{- 24 + \sqrt{{24}^{2} + 4 \cdot 20}}{- 40} , \frac{- 24 - \sqrt{{24}^{2} + 4 \cdot 20}}{- 40}\right]$

$y \in \left[- 0.04 , 1.24\right]$

The range is $y \in \left[- 0.04 , 1.24\right]$

graph{(x+6)/(x^2+5) [-10.47, 3.574, -3.903, 3.117]}