# How do you find the domain and range of y=(2x)/(-x-5)?

Feb 14, 2017

The domain of $y$ is $\mathbb{R} - \left\{- 5\right\}$
The range of $y$ is $\mathbb{R} - \left\{- 2\right\}$

#### Explanation:

As you cannot divide by $0$, $- x - 5 \ne 0$

The domain of $y$ is ${D}_{y} = \mathbb{R} - \left\{- 5\right\}$

To find the range, we need ${f}^{-} 1 \left(x\right)$

$y = 2 \frac{x}{-} x - 5$

$- y x - 5 y = 2 x$

$2 x + y x = - 5 y$

$x \left(y + 2\right) = - 5 y$

$x = \frac{- 5 y}{y + 2}$

Therefore,

${f}^{-} 1 \left(x\right) = \frac{- 5 x}{x + 2}$

The range of $f \left(x\right)$ is the domain of ${f}^{-} 1 \left(x\right)$

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