# What is the domain and range of  y = 3(x-2)/x?

Jan 31, 2017

The domain is $\mathbb{R} - \left\{0\right\}$
The range is $\mathbb{R} - \left\{3\right\}$

#### Explanation:

As you cannot divide by $0$, $\implies$, $x \ne 0$

The domain of $y$ is $\mathbb{R} - \left\{0\right\}$

To find the range, we need to calculate ${y}^{-} 1$

The domain of ${y}^{-} 1$ is the range

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

$y x = 3 x - 6$

$3 x - y x = 6$

$x \left(3 - y\right) = 6$

$x = \frac{6}{3 - y}$

Therefore,

${y}^{-} 1 = \frac{6}{3 - x}$

As you cannot divide by $0$, $\implies$, $x \ne 3$

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

graph{(y-(3x-6)/x)(y-3)(y-100x)=0 [-25.65, 25.65, -12.83, 12.82]}