Question

The domain of the function $f\left(x\right)=\sqrt{co{s}^{-1}\left(\begin{array}{c}\frac{1-|x|}{2}\end{array}\right)}$ is

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

Answer 1

Answer: (B)

$[-3,3]$

$f\left(x\right)=\sqrt{co{s}^{-1}\left(\begin{array}{c}\frac{1-|x|}{2}\end{array}\right)}$

for $f\left(x\right)$ to be defined ${\cos }^{-1}\left(\frac{1-|x|}{2}\right)\ge 0$

$\Rightarrow \frac{-\pi }{2}\le {\cos }^{-1}\left(\frac{1-|x|}{2}\right)\le \frac{\pi }{2}$

$\Rightarrow -1\le \frac{1-|x|}{2}\le 1$

$\Rightarrow -2\le 1-|x|\le 2$
$\Rightarrow -2-1\le -|x|\le 2-1$
$\Rightarrow -3\le -|x|\le 1$.
$\Rightarrow -1\le |x|\le 3$

$\Rightarrow |x|\ge -1$ but $|x|$ is always positive

$\Rightarrow |x|\le 3$

$\Rightarrow -3\le x\le 3$
$\Rightarrow x\in [-3,3]$