# How do you find the domain and range of  f(x) = (x+6 )/ (2x+1 )?

Jul 18, 2017

$f : \mathbb{R} \rightarrow \mathbb{R} , x \ne - \frac{1}{2}$

#### Explanation:

To find which values of $x$ don't belong to the domain of $f$, we have to set the denominator of $f = 0$ and solve for $x$

$2 x + 1 = 0$

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

Therefore, $- \frac{1}{2}$ is the only invalid real in the domain. Also, since $x = - \frac{1}{2}$ is an asymptote, we know that $f$ will increase with bound towards $\pm \infty$. Therefore, $f$ maps to all $y \in \mathbb{R}$.

Combining these two, we can define

$f : \mathbb{R} \rightarrow \mathbb{R}$