# What is the domain and range of (x+5)/(x^2+36)?

Nov 15, 2017

The domain is $x \in \mathbb{R}$.
The range is $y \in \left[- 0.04 , 0.18\right]$

#### Explanation:

The denominator is $> 0$

$\forall x \in \mathbb{R}$, ${x}^{2} + 36 > 0$

Therefore,

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

Let,

$y = \frac{x + 5}{{x}^{2} + 36}$

Simplifying and rearranging

$y \left({x}^{2} + 36\right) = x + 5$

$y {x}^{2} - x + 36 y - 5 = 0$

This is a quadratic equation in ${x}^{2}$

In order for this equation to have solutions, the discriminant $\Delta \ge 0$

So,

$\Delta = {b}^{2} - 4 a c = {\left(- 1\right)}^{2} - 4 \left(y\right) \left(36 y - 5\right) \ge 0$

$1 - 144 {y}^{2} + 20 y \ge 0$

$144 {y}^{2} - 20 y - 1 \le 0$

$y = \frac{20 \pm \sqrt{400 + 4 \cdot 144}}{288}$

${y}_{1} = \frac{20 + 31.24}{188} = 0.18$

${y}_{2} = \frac{20 - 31.24}{288} = - 0.04$

Therefore,

The range is $y \in \left[- 0.04 , 0.18\right]$

graph{(x+5)/(x^2+36) [-8.89, 8.884, -4.44, 4.44]}