# How do you find the domain and range of  y = (x + 3) / (x -5)?

Jun 9, 2017

The domain is $x \in \left(- \infty , 5\right) \cup \left(5 , + \infty\right)$
So, the range is $y \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$

#### Explanation:

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

The domain of $y$ is $x \in \left(- \infty , 5\right) \cup \left(5 , + \infty\right)$

To find the range, we need ${y}^{-} 1$

$y = \frac{x + 3}{x - 5}$

Interchange $y$ and $x$

$x = \frac{y + 3}{y - 5}$

Express $y$ in terms of $x$

$\left(y - 5\right) x = y + 3$

$x y - 5 x = y + 3$

$y \left(x - 1\right) = 5 x + 3$

$y = \frac{5 x + 3}{x - 1}$

This is ${y}^{-} 1$

The domain of ${y}^{-} 1$ is the range of $y$

So, the range is $y \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$