First slide

Functions (XII)

Question

The domain of definition of the function $f (x) = \sqrt{\sin^{- 1} (2 x) + \frac{π}{6}}$ for real-valued x is

Moderate

A

$[- \frac{1}{4}, \frac{1}{2}]$
B

$[- \frac{1}{2}, \frac{1}{2}]$
C

$(- \frac{1}{2}, \frac{1}{9})$
D

$[- \frac{1}{4}, \frac{1}{4}]$

Solution

$f (x) = \sqrt{\sin^{- 1} (2 x) + \frac{π}{6}} \sin^{- 1} (2 x) + \frac{π}{6} \geq 0 \Rightarrow \sin^{- 1} (2 x) \geq - \frac{π}{6} 2 x \geq \sin (- \frac{π}{6}) \Rightarrow 2 x \geq - \frac{1}{2} \Rightarrow x \geq - \frac{1}{4} a n d 2 x \leq 1 x \in [- \frac{1}{4}, \frac{1}{2}]$

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