Question

Find the domain of the following function:

$f\left(x\right)=\sqrt{1-\sqrt{1-\sqrt{1-x^2}}}$,
the domain is [-a, a], then a is?

Answer 1

$f\left(x\right)=\sqrt{1-\sqrt{1-\sqrt{1-x^2}}}$
$f\left(x\right)$ is defined when
(i) $1-x^2>0\Rightarrow -x^2>-1\Rightarrow x^2<1\Rightarrow x\in (-1,1)$

(ii) $1-\sqrt{1-x^2}\ge 0\Rightarrow -\sqrt{1-x^2}\ge -1$
$\Rightarrow \sqrt{1-x^2}\le 1\Rightarrow 1-x^2\le 1$
$\Rightarrow -x^2\le 0\Rightarrow x^2\ge 0\Rightarrow x\in R$

(iii) $1-\sqrt{1-\sqrt{1-x^2}}\ge 0$
$\Rightarrow +\sqrt{1-\sqrt{1-x^2}}\le 1\Rightarrow 1‘-\sqrt{1-x^2}\le 1$
$\Rightarrow 1-\sqrt{1-x^2}\le 1\Rightarrow -\sqrt{1-x^2}\le 0$
$\Rightarrow 1-\sqrt{1-x^2}>0\Rightarrow 1-x^2\ge 0$

From (i),(ii) and (iii), $x\in \{-1,1\}$