Question

Range of the function f defined by $f\left(x\right) =\left[\frac{1}{\sin\left\{x\right\}} \right]$ (where [. ] and {. } denote the greatest integer and the fractional part functions) is

• A. I and set of integers
• B. N, the set of natural numbers
• C. W, the set of whole numbers
• D. {2, 3, 4, .....}

Answer 1

Answer: (B)

$\because \left\{x\right\}\in [0,1)$
$ \therefore \left\{x\right\} \in [0, \sin1 )$ but $f(x)$ is defined if
$ \sin\left\{x\right\}\ne 0 $
$\therefore \frac{1}{\sin\left\{x\right\} } \in\left(\frac{1}{\sin1} , \infty\right) $
$\therefore \left[ \frac{1}{\sin\left\{x\right\}}\right] \in\left\{1, 2 , 3 ,....\right\} $