# What is the domain and range of  f(x)=sqrt[Cos[x]] ?

Sep 2, 2016

Domain:
$\left[- \frac{\pi}{2} + 2 \pi n , \frac{\pi}{2} + 2 \pi n\right]$ for all integer $n$

Range:
$\left[0 , 1\right]$

#### Explanation:

If author used square brackets inside $\cos$ just as brackets (not as an absolute value, which it looks like in a small font), than here is the explanation.

Also it should be noted that the square root in this problem used without $\pm$ sign implies arithmetic square root, that is only its non-negative value.

Square root requires $\cos x$ to be non-negative, which entails that $x$ must be in the intervals $\left[- \frac{\pi}{2} + 2 \pi n , \frac{\pi}{2} + 2 \pi n\right]$ for all integer $n$. That assures non-negativity of $\cos x$ and, therefore, constitutes the domain of this function.

In this domain $\cos x$ ranges in the interval $\left[0 , 1\right]$ and square root of it also will be in the same range.