Domain of definition of the function f(x) = sqrtsin^-1(2x) + (π)/6 for real valued x, is

Domain of definition of the function f(x) = sqrtsin^-1(2x)… — JEE Main 2003 Mathematics PYQ

AIEEE 2003

Domain of definition of the function

f(x) = \sqrt{\sin^{-1}(2x) + \frac{\pi}{6}}

for real valued

x

, is

A.
$\left[-\frac{1}{4}, \frac{1}{2}\right]$
B.
$\left[-\frac{1}{2}, \frac{1}{2}\right]$
C.
$\left(-\frac{1}{2}, \frac{1}{9}\right)$
D.
$\left[-\frac{1}{4}, \frac{1}{4}\right]$

Show answer and solution

Correct option: A

Solution

For the square root, we need sin⁻¹(2x) + π/6 ≥ 0, so sin⁻¹(2x) ≥ -π/6, meaning 2x ≥ sin(-π/6) = -1/2, so x ≥ -1/4.

Also, sin⁻¹(2x) is defined only when -1 ≤ 2x ≤ 1, i.e., -1/2 ≤ x ≤ 1/2.

Combining both conditions: x ∈ [-1/4, 1/2].

[AIEEE 2003]

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