Question

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

• A. $Z,$ the 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)

$N,$ the set of natural numbers

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