# What is the domain and range of f(x) = (x+6 )/ (2x+1) ?

Jan 16, 2018

The domain is $x \in \mathbb{R} - \left\{- \frac{1}{2}\right\}$.
The range is $y \in \mathbb{R} - \left\{\frac{1}{2}\right\}$

#### Explanation:

As you cannot divide by $0$, the denominator is $\ne 0$

Therefore,

$2 x + 1 \ne 0$

$\implies$, x"=-1/2

The domain is $x \in \mathbb{R} - \left\{- \frac{1}{2}\right\}$

In order to find the range, proceed as follows .

Let $y = \frac{x + 6}{2 x + 1}$

$y \left(2 x + 1\right) = x + 6$

$2 x y + y = x + 6$

$2 x y - x = 6 - y$

$x \left(2 y - 1\right) = \left(6 - y\right)$

$x = \frac{6 - y}{2 y - 1}$

In order for $x$ to have solutions,

$2 y - 1 \ne 0$

$y \ne \frac{1}{2}$

The range is $y \in \mathbb{R} - \left\{\frac{1}{2}\right\}$

graph{(x+6)/(2x+1) [-18.02, 18.01, -9.01, 9.01]}