Find the domain of the following function:f(x)= ^ -1 x^2-1 - Vidyadip

Question

Find the domain of the following function:$\text{f(x)}=\sin^{-1}\sqrt{\text{x}^2-1}$

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Answer

To the domain of $\sin^{-1}y$ which is $[-1, 1]$
$\therefore$ $x^2 -1$ $\in$ $[0, 1]$ as square root can not be negative$\Rightarrow x^2$ $\in$ $[1, 2]$
$\Rightarrow\text{x}\in\big[-\sqrt2,-1\big]\cup\big[1,\sqrt2\big]$ Hence, the domain is $\big[-\sqrt2,-1\big]\cup\big[1,\sqrt2\big]$

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