# How do you find the domain of f(x) = (x+18 )/( x^2-169)?

Sep 28, 2017

$x \in \mathbb{R} , x \ne \pm 13$

#### Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be.

$\text{solve } {x}^{2} - 169 = 0 \Rightarrow \left(x - 13\right) \left(x + 13\right) = 0$

$\Rightarrow x = \pm 13 \leftarrow \textcolor{red}{\text{ excluded values}}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne \pm 13$

Sep 28, 2017

The domain is $x \in \left(- \infty , - 13\right) \cup \left(- 13 , + 13\right) \cup \left(+ 13 , + \infty\right)$

#### Explanation:

The denominator must $\ne 0$

Therefore,

${x}^{2} - 169 \ne 0$

$\left(x + 13\right) \left(x - 13\right) \ne 0$

Let $g \left(x\right) = \left(x + 13\right) \left(x - 13\right)$

Construct a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a}$$- 13$$\textcolor{w h i t e}{a a a a a a a}$$13$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 13$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 13$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$color(white)(aaaaa)-$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$g \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a}$$+$

Therefore,

$g \left(x\right) \ne 0$ when $x \in \left(- \infty , - 13\right) \cup \left(- 13 , + 13\right) \cup \left(+ 13 , + \infty\right)$

This is the domain of $f \left(x\right)$

graph{(x+18)/(x^2-169) [-20.27, 20.28, -10.14, 10.14]}