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

Dec 23, 2016

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

#### Explanation:

As you cannot divide by $0$, $x \ne \frac{1}{2}$

There is a vertical asymptote $x = \frac{1}{2}$

Therefore,

The domain of $f \left(x\right)$ is ${D}_{f \left(x\right)}$ is $x \in \mathbb{R} - \left\{\frac{1}{2}\right\}$

${\lim}_{x \to \pm \infty} f \left(x\right) = {\lim}_{x \to \pm \infty} \frac{2 x}{2} x = 1$

There is a horizontal asymptote $y = 1$

The range is $f \left(x\right) \in \mathbb{R} - \left\{1\right\}$

graph{(y-(2x+1)/(2x-1))(y-1)(x-1/2)=0 [-12.66, 12.65, -6.33, 6.33]}