Question

Find the domain of definations of the following functions:
$f\left(x\right)=\sqrt{1-\sqrt{1-x^2}}$

Answer 1

$\begin{array}{c}Fordomain\\ 1-x^2\ge 0\\ \Rightarrow \left(1-x\right)\left(1+x\right)\ge 0\\ \Rightarrow \left(1-x\right)\left(1+x\right)\le 0\\ \Rightarrow x\in \left[-1,1\right]\\ also\\ 1-\sqrt{1-x^2}\ge 0\\ \Rightarrow -\sqrt{1-x^2}\ge -1\\ \Rightarrow \sqrt{1-x^2}\le 1\\ \Rightarrow 1-x^2\le 1\\ \Rightarrow -x^2\le 0\\ \Rightarrow x^2\ge 0\end{array}$

Hence domain of $f\left(x\right)$ is $[-1,1]$