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

Jun 26, 2018

The domain is $= \mathbb{R}$.
The range is $y \in \left[- 0.4 , 1.24\right]$

#### Explanation:

The function is

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

$\forall x \in \mathbb{R}$, the denominator is ${x}^{2} + 5 > 0$

The domain is $= \mathbb{R}$

To calculate the range, let

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

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

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

This is a quadratic equation in ${x}^{2}$ and in order to have solutions, the discriminant must be $\ge 0$

Therefore,

$\Delta = {\left(- 1\right)}^{2} - 4 \left(y\right) \left(5 y - 6\right) \ge 0$

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

$\implies$, $20 {y}^{2} - 24 y - 1 \le 0$

The solutions to this inequality is

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

$y \in \left[- 0.4 , 1.24\right]$

The range is $y \in \left[- 0.4 , 1.24\right]$

graph{(x+6)/(x^2+5) [-13.7, 14.78, -7.27, 6.97]}