Domain of definition of the function f ( x ) = sin - 1 ( 2 x ) + 6 for real valued x , is [2003]

Question

Domain of definition of the function $f (x) = \sqrt{\sin^{- 1} (2 x) + \frac{π}{6}}$ for real valued $x$ , is [2003]

Smart Achievers · Accepted Answer

(1)

For $f (x) = \sqrt{\sin^{- 1} (2 x) + \frac{π}{6}}$ to be defined and real, $\sin^{- 1} (2 x) + \frac{π}{6} \geq 0$

$\Rightarrow \sin^{- 1} (2 x) \geq - \frac{π}{6}$ $\dots (i)$

But, $- \frac{π}{2} \leq \sin^{- 1} (2 x) \leq \frac{π}{2}$ $\dots (i i)$

On combining (i) and (ii), we get

$- \frac{π}{6} \leq \sin^{- 1} (2 x) \leq \frac{π}{2}$

$\Rightarrow \sin (- \frac{π}{6}) \leq 2 x \leq \sin (\frac{π}{2})$

$\Rightarrow - \frac{1}{2} \leq 2 x \leq 1$

$\Rightarrow - \frac{1}{4} \leq x \leq \frac{1}{2}$

$∴$ $Domain = [- \frac{1}{4}, \frac{1}{2}]$

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