# What is the domain and range of f(x) = (x-4) /( x+2)??

Oct 7, 2015

Domain: $\mathbb{R} - \left\{- 2\right\}$ or (equivalently) $\mathbb{R} | \left(- \infty , - 2\right) \cup \left(- 2 , + \infty\right)$

Range: $\mathbb{R} - \left\{1\right\}$

#### Explanation:

$f \left(x\right) = \frac{x - 4}{x + 2}$ is defined for all Real values of $x$ except when $\left(x + 2\right) = 0$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow$ except for $x = - 2$
So the Domain is all $\mathbb{R}$ except $\left(- 2\right)$

To find the Range consider $\overline{f} \left(x\right)$, the inverse of $f \left(x\right)$

By definition of inverse
$\textcolor{w h i t e}{\text{XXX}} f \left(\overline{f} \left(x\right)\right) = x$
and therefore
$\textcolor{w h i t e}{\text{XXX}} \frac{\overline{f} \left(x\right) - 4}{\overline{f} \left(x\right) + 2} = x$

$\textcolor{w h i t e}{\text{XXX}} \overline{f} \left(x\right) - 4 = x \cdot \overline{f} \left(x\right) + 2 x$

$\textcolor{w h i t e}{\text{XXX}} \overline{f} \left(x\right) - x \overline{f} \left(x\right) = 2 x + 4$

$\textcolor{w h i t e}{\text{XXX}} \overline{f} \left(x\right) \left(1 - x\right) = 2 x + 4$

$\textcolor{w h i t e}{\text{XXX}} \overline{f} \left(x\right) = \frac{2 x + 4}{1 - x}$

Which is defined for all values of $x \ne 1$

That is the Domain of $\overline{f} \left(x\right)$ is $\mathbb{R} - \left\{1\right\}$
and
Since the Domain of a function is the Range of its inverse
$\textcolor{w h i t e}{\text{XXX}}$the Range of $f \left(x\right)$ is $\mathbb{R} - \left\{1\right\}$