Find the domain of f(x)=2 ^ -1 2x= ^ -1 x. - Vidyadip

Question

Find the domain of $\text{f(x)}=2\cos^{-1}2\text{x}=\sin^{-1}\text{x}.$

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Answer

Domain of $\cos^{-1}\text{x}$ lies in the interval [-1, 1]
$\therefore$ Domain of $\cos^{-1}(2\text{x})$ lies in the interval [-1, 1]
$\Rightarrow-1\leq2\text{x}\leq1$
$\Rightarrow\frac{-1}{2}\leq\text{x}\leq\frac{1}{2}$
Domain of $\cos^{-1}(2\text{x})$ is $\Big[-\frac{1}{2},\frac{1}{2}\Big]$
Domain of $\cos^{-1}\text{x}$ lies in the interval [-1, 1]
$\therefore$ Domain of $\cos^{-1}(2\text{x})+\sin^{-1}\text{x}$ lies in the interval $\Big[-\frac{1}{2},\frac{1}{2}\Big].$

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