# How do you find the domain and range of 3-cos2x?

Aug 5, 2017

Domain:$\left(- \infty , + \infty\right)$
Range: $\left[4 , 2\right]$

#### Explanation:

$f \left(x\right) = 3 - \cos 2 x$

$f \left(x\right)$ is defined $\forall x \in \mathbb{R}$

Hence, the domain of $f \left(x\right) = \left(- \infty , + \infty\right)$

Let $\theta = 2 x$

$\therefore f \left(x\right) = 3 - \cos \theta$

The range of $\cos \theta = \pm 1$

$\therefore$ the range of $f \left(x\right) = \left[3 + 1 , 3 - 1\right] = \left[4 , 2\right]$

This can be see from the graph of $f \left(x\right)$ below.

graph{3-cos(2x) [-10.21, 9.79, -1.76, 8.24]}