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

Apr 4, 2017

Domain is {x in RR ; x != -5 }
Range is {y in RR ; y != 0 }

#### Explanation:

Domain: Denominator should not be $0 \therefore x + 5 \ne 0 \mathmr{and} x \ne - 5$
Domain is any real value except $x = - 5$ or {x in RR ; x != -5 }

Range is any real value except $y = 0$ or {y in RR ; y != 0 } graph{1/(x+5) [-10, 10, -5, 5]}

Apr 4, 2017

see explanation.

#### Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be.

$\text{solve } x + 5 = 0 \Rightarrow x = - 5$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne - 5$

$\text{Rearrange "y " to make x the subject}$

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

$\Rightarrow x y + 5 y = 1 \Rightarrow x y = 1 - 5 y$

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

Applying the same reasoning as for the domain we obtain.

$\text{range is } y \in \mathbb{R} , y \ne 0$