# What is the range of the function f(x)=2/(x+3) -4?

Apr 30, 2017

$y \in \mathbb{R} , y \ne - 4$

#### Explanation:

$\text{Rearrange f(x) to make x the subject}$

$y = f \left(x\right) = \frac{2}{x + 3} - \frac{4 \left(x + 3\right)}{x + 3}$

$\Rightarrow y = \frac{2 - 4 x - 12}{x + 3} = \frac{- 4 x - 10}{x + 3}$

$\textcolor{b l u e}{\text{cross-multiply}}$

$\Rightarrow y x + 3 y = - 4 x - 10$

$\Rightarrow y x + 4 x = - 10 - 3 y$

$\Rightarrow x \left(y + 4\right) = - 10 - 3 y$

$\Rightarrow x = \frac{- 10 - 3 y}{y + 4}$

The denominator cannot be zero as this would make the function $\textcolor{b l u e}{\text{undefined}} .$Equating the denominator to zero and solving gives the value that y cannot be.

$\text{solve "y+4=0rArry=-4larrcolor(red)" excluded value}$

$\text{range } y \in \mathbb{R} , y \ne - 4$