# How do you find domain and range for f(x) = (4x - 7) /( 6 - 5x)?

Jul 2, 2018

The domain is $x \in \left(- \infty , \frac{6}{5}\right) \cup \left(\frac{6}{5} , + \infty\right)$.
The range is $y \in \left(- \infty , - \frac{4}{5}\right) \cup \left(- \frac{4}{5} , + \infty\right)$

#### Explanation:

The denominator must be $\ne 0$

$\implies$, $6 - 5 x \ne 0$

$\implies$, $x \ne \frac{6}{5}$

The domain is $x \in \left(- \infty , \frac{6}{5}\right) \cup \left(\frac{6}{5} , + \infty\right)$

To find the range, let

$y = \frac{4 x - 7}{6 - 5 x}$

$\implies$, $y \left(6 - 5 x\right) = 4 x - 7$

$\implies$, $6 y - 5 x y = 4 x - 7$

$\implies$, $4 x + 5 x y = 6 y + 7$

$\implies$, $x \left(4 + 5 y\right) = 6 y + 7$

$\implies$, $x = \frac{6 y + 7}{5 y + 4}$

The denominator must be $\ne 0$

$\implies$, $5 y + 4 \ne 0$

$\implies$, $y \ne - \frac{4}{5}$

The range is $y \in \left(- \infty , - \frac{4}{5}\right) \cup \left(- \frac{4}{5} , + \infty\right)$

graph{(4x-7)/(6-5x) [-11.25, 11.25, -5.625, 5.625]}