# What is the domain and range of f(x)=2sin(x+pi)-4?

Dec 24, 2015

Domain: $\mathbb{R}$
Range: $\left[- 6 , - 2\right] \in \mathbb{R}$

#### Explanation:

$\sin \left(x + \pi\right)$ is defined for all Real values of $x$
therefore $2 \sin \left(x + \pi\right) - 4$ is defined for all real values of $x$
$\Rightarrow$ Domain $= \mathbb{R}$

$\sin \left(\theta\right)$ for all $\theta \in \mathbb{R}$ (including $\theta = x + \pi$) has a Range of $\left[- 1 , + 1\right]$
therefore $2 \sin \left(x + \pi\right)$ has a Range of $\left[- 2 , + 2\right]$
and
$2 \sin \left(x + \pi\right) - 4$ has a Range of $\left[- 2 - 4 , + 2 - 4\right] = \left[- 6 , - 2\right]$ (in $\mathbb{R}$)