# What is the domain and range of y=(x^2 -5x -6) / (x^2 -3x -18)?

Mar 17, 2018

The domain of the function is $x \in \mathbb{R} - \left\{- 3\right\}$. The range is $y \in \mathbb{R} - \left\{1\right\}$

#### Explanation:

Factorise the numerator and denominator

$y = \frac{{x}^{2} - 5 x - 6}{{x}^{2} - 3 x - 18} = \frac{\left(x + 1\right) \cancel{x - 6}}{\left(x + 3\right) \cancel{x - 6}}$

$= \frac{x + 1}{x + 3}$

The denominator is $\ne 0$, therefore

$x + 3 \ne 0$, $\implies$, $x \ne - 3$

The domain of the function is x in RR-{-3}

To determine the range, proceed as follows

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

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

$y x - x = 1 - 3 y$

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

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

The denominator is $\ne 0$

$y - 1 \ne 0$, $\implies$, $y \ne 1$

The range is $y \in \mathbb{R} - \left\{1\right\}$

graph{(x^2-5x-6)/(x^2-3x-18) [-16.02, 16.02, -8.01, 8.01]}