# How do you find the domain and range for f(x)=(2x+1)/(x-3)?

Apr 29, 2017

The domain of $= \mathbb{R} - \left\{3\right\}$
The range of $= \mathbb{R} - \left\{2\right\}$

#### Explanation:

As we cannot divide by $0$, $x \ne 3$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{3\right\}$

Let $y = \frac{2 x + 1}{x - 3}$

Then,

$y x - 3 y = 2 x + 1$

$y x - 2 x = 3 y + 1$

$x \left(y - 2\right) = 3 y + 1$

$x = \frac{3 y + 1}{y - 2}$

Therefore,

${f}^{-} 1 \left(x\right) = \frac{3 x + 1}{x - 2}$

The domain of $x$ is the range of $y$

The range of $f \left(x\right)$ is ${R}_{f} \left(x\right) = \mathbb{R} - \left\{2\right\}$
graph{(2x+1)/(x-3) [-28.86, 28.9, -14.43, 14.43]}