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)

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