# How do you find the domain and range of f(x)= 1/(x+1)?

Feb 14, 2017

The domain is $= \mathbb{R} - \left\{- 1\right\}$
The range is $= \mathbb{R} - \left\{0\right\}$

#### Explanation:

As you cannot divide by $0$, $x \ne - 1$

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

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

Let $y = \frac{1}{x + 1}$

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

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

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

Therefore,

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

The domain of ${f}^{-} 1 \left(x\right)$ is ${D}_{f}^{-} 1 \left(x\right) = \mathbb{R} - \left\{0\right\}$

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

The range is ${R}_{y} = \mathbb{R} - \left\{0\right\}$