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Question: Domain of \(f ( x ) = \sqrt { \cos ( \sin x ) } + \sqrt { \log _ { x } \{ x \} }\) ; where {.}denot...

Domain of f(x)=cos⁡(sin⁡x)+log⁡x{x}f ( x ) = \sqrt { \cos ( \sin x ) } + \sqrt { \log _ { x } \{ x \} } ; where {.}denotes the fractional part, is

A

[1, π)

B

(0, 2π) ~ [1, π)

C

(0,π2)∼{1}\left( 0 , \frac { \pi } { 2 } \right) \sim \{ 1 \}

D

(0, 1)

Answer

(0, 1)

Explanation

Solution

We must have cos(sinx) > 0

⇒ π/2 > sin x ≥ −π/2, which is true for all real x. secondly logx{x} > 0.

If 0 < x < 1 ⇒ 0 < {x} < 1 ⇒ 0 < x < 1

If x > 1 ⇒ {x} ≥ 1, which is not possible. Thus domain is

(0, 1).