The domain of the function f (x) = 4 ^ -1 ((1 - |x|)/(2)) is

$\cos^{- 1}$\left(\dfrac{1 - \left|x\right|}{2}\right) is defined if

$- 1 \le \dfrac{1 - \left|x\right|}{2} \le 1$
$⇒ - 2 \le 1 - \left|x\right| \le 2 ⇒ - 3 \le - \left|x\right| \le 1$
$⇒ - 1 \le \left|x\right| \le 3$
\left|x\right| \ge - 1$is true for all real values of$x$.<br />\left|x\right| \le 3 ⇒ - 3 \le x \le 3$
Also, $4\sqrt{\cos^{- 1} \left(\dfrac{1 - \left|x\right|}{2}\right)}$ is defined if

$\cos^{- 1} \left(\dfrac{1 - \left|x\right|}{2}\right) \ge 0$
$⇒ \dfrac{1 - \left|x\right|}{2} \ge \cos 0 = 1$
$⇒ 1 - \left|x\right| \ge 2$
$⇒ \left|x\right| \le - 1$ (not possible)
$∴$ Domain of $f\left(x\right) = \left[- 3,3\right]$