# How do you find the domain and range of f(x)=2/(x-1)?

Jul 13, 2018

The domain is $x \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$. The range is $y \in \left(- \infty , 0\right) \cup \left(0 , + \infty\right)$.

#### Explanation:

The function is

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

As we cannot divide by $0$, the denominator must be $\ne 0$

$x - 1 \ne 0$

$x \ne 1$

The domain is $x \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$

To find the range, let

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

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

$y x - y = 2$

$y x = 2 + y$

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

As we cannot divide by $0$, the denominator must be $\ne 0$

$y \ne 0$

The range is $y \in \left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

graph{2/(x-1) [-12.66, 12.65, -6.33, 6.33]}