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

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Solution

For $2 \cos^{- 1} 2 x$ to be defined.
$- 1 \leq 2 x \leq 1 \Rightarrow - \frac{1}{2} \leq x \leq \frac{1}{2} . . . . . (i)$
For $\sin^{- 1} x$ to be defined.
$- 1 \leq x \leq 1 . . . . . (ii)$
Domain of $f (x) = [- \frac{1}{2}, \frac{1}{2}] \cap [- 1, 1]$
$= [- \frac{1}{2}, \frac{1}{2}]$ .

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f (x) = 2 {cos}^{- 1} 2 x + {sin}^{- 1} x

2 c o s^{- 1} x = s i n^{- 1} (2 x \sqrt{1 - x^{2}})

2 {cos}^{- 1} x = {sin}^{- 1} (2 x \sqrt{1 - x^{2}})

{sin}^{- 1} (2 x \sqrt{1 - x^{2}}) = 2 {cos}^{- 1} x, \frac{1}{\sqrt{2}} \leq x \leq 1

2 {cos}^{- 1} x = {sin}^{- 1} (2 x \sqrt{1 - x^{2}})

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