Question

The domain of the function $f\left(x\right)=\sqrt{{\cos }^{-1}\left(\frac{1-\left|x\right|}{2}\right)}$ is

• A. $(-3,3)$
• B. $[-3,3]$
• C. $[0,3]$
• D. $(-\infty ,-3)\cup (3,\infty )$

Answer 1

The correct option is B $[-3,3]$

$f\left(x\right)=\sqrt{{\cos }^{-1}\left(\frac{1-\left|x\right|}{2}\right)}$
As we know that ${\cos }^{-1}\left(\frac{1-\left|x\right|}{2}\right)\in [0,\pi ]$ which is non-negative.

Now, $-1\le \frac{1-\left|x\right|}{2}\le 1$
$\Rightarrow -2\le 1-|x|\le 2$
$\Rightarrow -3\le -|x|\le 1$
$\Rightarrow -1\le |x|\le 3$
$\Rightarrow 0\le |x|\le 3(\because |x|\ge 0)$
$\Rightarrow x\in [-3,3]$