# What is the range of the function f(x) = 1 / (x-2)?

May 14, 2017

The range of $f \left(x\right)$ is $= \mathbb{R} - \left\{0\right\}$

#### Explanation:

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

Here,

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

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

Interchanging $x$ and $y$

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

Solving for $y$

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

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

Therefore,

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

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

Therefore,

The range of $f \left(x\right)$ is $= \mathbb{R} - \left\{0\right\}$
graph{1/(x-2) [-12.66, 12.65, -6.33, 6.33]}