# What is the domain and range of g(x) = (7x+4)/(x+4)?

Apr 29, 2018

The domain is $x \in \left(- \infty , - 4\right) \cup \left(- 4 , + \infty\right)$.
The range is $y \in \left(- \infty , 7\right) \cup \left(7 , + \infty\right)$

#### Explanation:

Let $y = \frac{7 x + 4}{x + 4}$

The denominator is $\ne 0$

Therefore,

$x + 4 \ne 0$

$x \ne - 4$

The domain is $x \in \left(- \infty , - 4\right) \cup \left(- 4 , + \infty\right)$

Also,

$y \left(x + 4\right) = 7 x + 4$

$y x + 4 y = 7 x + 4$

$y x - 7 x = 4 - 4 y$

$x \left(y - 7\right) = \left(4 - 4 y\right)$

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

The denominator is $\ne 0$

Therefore

$y - 7 \ne 0$

$y \ne 7$

The range is $y \in \left(- \infty , 7\right) \cup \left(7 , + \infty\right)$

graph{(y-((7x+4)/(x+4)))(y-7)=0 [-36.53, 36.52, -18.28, 18.27]}