Correct Answer - Option 3 :
\(\left[ { - \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right]\)
Concept:
The domain of inverse sine function, sin x is \(x \in \left[ { - 1,1} \right]\)
Calculation:
Domain of the function is calculated as follows:
\({\sin ^{ - 1}}\left( {2x\sqrt {1 - {x^2}} } \right)\)
\(- 1 \le 2x\sqrt {1 - {x^2}} \le 1\)
\( - \frac{1}{2} \le x\sqrt {1 - {x^2}} \le \frac{1}{2}\)
\({x^2}\left( {1 - {x^2}} \right) \le \frac{1}{4}\)
\(t - {t^2} - \frac{1}{4} \le 0\)
\({\left( {t - \frac{1}{2}} \right)^2} \le 0\)
\(t \le \frac{1}{2}\)
\({x^2} \le \frac{1}{{\sqrt 2 }}\)
\(x \in \left[ { - \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right]\)
