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Question: Domain of definition of the function \(f(x) = \sqrt{\sin^{- 1}(2x) + \frac{\pi}{6}},\) for real valu...

Domain of definition of the function f(x)=sin⁡−1(2x)+π6,f(x) = \sqrt{\sin^{- 1}(2x) + \frac{\pi}{6}}, for real value x,x, is

A

[−14,12]\left\lbrack - \frac{1}{4},\frac{1}{2} \right\rbrack

B

[−12,12]\left\lbrack - \frac{1}{2},\frac{1}{2} \right\rbrack

C

(−12,19)\left( - \frac{1}{2},\frac{1}{9} \right)

D

[−14,14]\left\lbrack - \frac{1}{4},\frac{1}{4} \right\rbrack

Answer

[−14,12]\left\lbrack - \frac{1}{4},\frac{1}{2} \right\rbrack

Explanation

Solution

−π6≤sin⁡−1(2x)≤π2⇒−12≤2x≤1- \frac{\pi}{6} \leq \sin^{- 1}(2x) \leq \frac{\pi}{2} \Rightarrow - \frac{1}{2} \leq 2x \leq 1 ⇒ x∈[−14,12]x \in \left\lbrack - \frac{1}{4},\frac{1}{2} \right\rbrack.