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

Apr 27, 2017

The domain of $f \left(x\right)$ is $\mathbb{R} - \left\{2\right\}$
The range of $f \left(x\right)$ is $\mathbb{R} - \left\{1\right\}$

#### Explanation:

As we cannot divide by $0$, $x \ne 2$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{2\right\}$

Let $y = \frac{x + 2}{x - 2}$

Then,

$y x - 2 y = x + 2$

$y x - x = 2 y + 2$

$x \left(y - 1\right) = 2 \left(y + 1\right)$

$x = 2 \frac{y + 1}{y - 1}$

Therefore,

f^-1(x))=2(x+1)/(x-1)

The domain of $x$ is the range of $y$

The range of $f \left(x\right)$ is ${R}_{f} \left(x\right) = \mathbb{R} - \left\{1\right\}$

graph{(y-(x+2)/(x-2))(y-1)=0 [-9.25, 10.75, -2.93, 7.07]}