# How do you find the domain and range of 1/(x+3)+3?

Mar 7, 2018

The domain is $x \in \mathbb{R} - \left\{- 3\right\}$. The range is $\mathbb{R} - \left\{3\right\}$

#### Explanation:

Let

$y = \frac{1}{x + 3} + 3 = \frac{1 + 3 \left(x + 3\right)}{x + 3} = \frac{1 + 3 x + 9}{x + 3}$

$= \frac{3 x + 10}{x + 3}$

As you cannot divide by $0$, the

$\text{denominator } \ne 0$

$x + 3 \ne 0$

$x \ne - 3$

Therefore,

The domain is $x \in \mathbb{R} - \left\{- 3\right\}$

To find the range, proceed as follows

$y = \frac{3 x + 10}{x + 3}$

$y \left(x + 3\right) = 3 x + 10$

$y x + 3 y = 3 x + 10$

$y x - 3 x = 10 - 3 y$

$x \left(y - 3\right) = \left(10 - 3 y\right)$

$x = \frac{10 - 3 y}{y - 3}$

AS you cannot divide by $0$,

$y - 3 \ne 0$

$y \ne 3$

The range is $\mathbb{R} - \left\{3\right\}$

graph{(3x+10)/(x+3) [-18.67, 17.36, -5.59, 12.43]}