Question

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

• A. $\left[-1,-\frac{1}{\sqrt{2}}\right]\cup \left[\frac{1}{\sqrt{2}},1\right]$
• B. $\left[-1,1\right]$
• C. $(-\infty ,-\frac{1}{2}]\cup [\frac{1}{\sqrt{2}},\infty )$
• D. $\left[\frac{1}{\sqrt{2}},1\right]$

Answer 1

The correct option is D $\left[\frac{1}{\sqrt{2}},1\right]$

$f\left(x\right)=\sqrt{x-\sqrt{1-x^2}}$
$fordomain$
$x-\sqrt{1-x^2}\ge 0$
$\Rightarrow x\ge \sqrt{1-x^2}$
$squaringbothsides$
$x^2\ge 1-x^2$
$\Rightarrow 2x^2-1\ge 0$
$\Rightarrow 2x^2\ge 1$
$\Rightarrow x^2\ge \frac{1}{2}$
$\Rightarrow x\ge \pm \frac{1}{\sqrt{2}}$
$rejectingthenegativeterm$
$Domain:[\frac{1}{\sqrt{2}},1]$