First slide

Functions (XII)

Question

The domain of the function
$f (x) = \frac{1}{\sqrt{\{\sin x\} + \{\sin (π + x)\}}}$ where $\{\}$ denotes the fractional part, is

Moderate

A

$[0, π]$
B

$(2 n + 1) \frac{π}{2}, n \in Z$
C

$(0, π)$
D

$R - \{\frac{n π}{2}, n \in Z\}$

Solution

$f (x) = \frac{1}{\sqrt{\{\sin x\} + \{\sin (π + x)\}}}$
$= \frac{1}{\sqrt{\{\sin x\} + \{- \sin x\}}}$
Now
$\{\sin x\} + \{- \sin x\} = \{\begin{matrix} 0 sinx is an integer \\ 1 \sin x is not an integer \end{matrix}$
For f(x) to get defined
$\{\sin x\} + \{- \sin x\} \neq 0$
$\Rightarrow \sin x \neq$ integer
$\Rightarrow \sin x \neq \pm 1, 0 \Rightarrow x \neq \frac{n π}{2}, n \in I$
Hence, the domain is $R - \{\frac{n π}{2} / n \in I\}$

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