# How do you find the domain and range of (x - 2 )/ (x^2- 16)?

Oct 12, 2017

Domain : $x \in \mathbb{R} | x \ne - 4 , x \ne 4$ ,in interval notation , $\left(- \infty , - 4\right) \cup \left(- 4 , 4\right) \cup \left(4 , \infty\right)$
Range : $f \left(x\right) \in \mathbb{R} \mathmr{and} \left(- \infty , \infty\right)$

#### Explanation:

$f \left(x\right) = \frac{x - 2}{{x}^{2} - 16} = \frac{x - 2}{\left(x + 4\right) \left(x - 4\right)}$ .

Domain (input $x$) : Denominator should not be zero ,otherwise

$f \left(x\right)$ will be undefined $\therefore x + 4 \ne 0 \mathmr{and} x \ne - 4$ and

$x - 4 \ne 0 \mathmr{and} x \ne 4$, so $x$ can be any real number except

$x = 4 \mathmr{and} x = - 4$

Domain : $x \in \mathbb{R} | x \ne - 4 , x \ne 4$ or

in interval notation , $\left(- \infty , - 4\right) \cup \left(- 4 , 4\right) \cup \left(4 , \infty\right)$

Range : $f \left(x\right)$ can be any real number

Range : $f \left(x\right) \in \mathbb{R} \mathmr{and} \left(- \infty , \infty\right)$

graph{(x-2)/(x^2-16) [-10, 10, -5, 5]}