Question

The domain of the function $f(x)=\sqrt{\cos x}$ is

• A. $\left[0,\frac{\pi }{2}\right]$
• B. $\left[0,\frac{\pi }{2}\right]\cup \left[\frac{3\pi }{2},2\pi \right]$
• C. $\left[\frac{3\pi }{2},2\pi \right]$
• D. $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$

Answer 1

Answer: (B), (C), (D)

Given, $f(x)=\sqrt{\cos x}$
i.e., $\cos x\ge 0$
But $-1\le \cos x\le 1$
$\therefore 0\le \cos x\le 1$
i.e., $x$ lies in Ist or IVth quadrant
$\Rightarrow 0\le x\le \frac{\pi }{2}$ or $\frac{3\pi }{2}\le x\le 2\pi$
$\therefore x\in \left[0,\frac{\pi }{2}\right]\cup \left[\frac{3\pi }{2},2\pi \right]$
Also, $\cos (-x)=\cos x$
Hence, $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ is also the domain of the function.