# How do you find the domain and range of y= .5cos(x/2-2sqrt3)+1?

Sep 10, 2017

The domain is $x \in \mathbb{R}$
The range is $y \in \left[0.5 , 1.5\right]$

#### Explanation:

By definition, cosine is defined over $\mathbb{R}$

so, the domain is $= \mathbb{R}$

The range of cosine is

$- 1 \le \cos \left(\frac{x}{2} - 2 \sqrt{3}\right) \le 1$

Multiply by $0.5$

$- 1 \cdot 0.5 \le 0.5 \cdot \cos \left(\frac{x}{2} - 2 \sqrt{3}\right) \le 1 \cdot 0.5$

$- 0.5 \le 0.5 \cdot \cos \left(\frac{x}{2} - 2 \sqrt{3}\right) \le 0.5$

Add $1$

$- 0.5 + 1 \le 0.5 \cdot \cos \left(\frac{x}{2} - 2 \sqrt{3}\right) + 1 \le 0.5 + 1$

$+ 0.5 \le 0.5 \cdot \cos \left(\frac{x}{2} - 2 \sqrt{3}\right) + 1 \le 1.5$

$+ 0.5 \le y \le 1.5$