# How do you find the domain and range of f(x) = 1 / (x-5)?

Feb 4, 2017

The domain is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{5\right\}$
The range is ${R}_{f} \left(x\right) = \mathbb{R} - \left\{0\right\}$

#### Explanation:

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

So, the domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{5\right\}$

Let $y = \frac{1}{x - 5}$

$y \left(x - 5\right) = 1$

$y x - 5 y = 1$

$y x = 1 + 5 y$

$x = \frac{1 + 5 y}{y}$

Therefore,

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

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

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 ${R}_{f \left(x\right)} = \mathbb{R} - \left\{0\right\}$