# What is the domain and range of -3 cos x?

Feb 22, 2018

Domain: $\left(- \infty , \infty\right)$; Range: $\left[- 3 , 3\right]$

#### Explanation:

The range of a function is the set of values between the maximum and minimum values that a certain function can output for its domain. The domain of a function is all the function input values that will produce a valid output value.

The standard range of $\cos \left(\theta\right) = \left[- 1 , 1\right]$ and the domain: $\left(- \infty , \infty\right)$. In other words, the maximum value of $\cos \left(\theta\right)$ is 1, the minimum value is -1, and all numbers can be plugged into $\cos \left(\theta\right)$ and a valid function value will be found. By multiplying $\cos \left(\theta\right)$ by $- 3$, all values of $\cos \left(\theta\right)$ will be inverted because of the negative sign, and all values will be increased by a factor of 3 as well.

$\cos \left(0\right) = 1$ (maximum value)
$- 3 \left(\cos \left(0\right)\right) = - 3 \left(1\right) = - 3$ (new minimum value)

Multiplying $\cos \left(\theta\right)$ by a factor of -3 does not affect the set of valid numbers that can be entered into $\cos \left(\theta\right)$ where a valid function value can be found, so the domain of $- 3 \cos \left(\theta\right)$ is the same as $\cos \left(\theta\right)$.