First slide

Functions (XII)

Question

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

Moderate

A

I, the set of integers
B

N, the set of natural numbers
C

W, the set of whole numbers
D

{1,2,3,4,...}

Solution

Since $\{x\} \in [0, 1), \sin \{x\} \in (0, \sin 1)$ as f(x) is defined if $\sin \{x\} \neq 0$ i.e.,
$\frac{1}{\sin \{x\}} \in (\frac{1}{\sin 1}, \infty) or [\frac{1}{\sin \{x\}}] \in \{1, 2, 3, . . . . . .\}$
Note that, $1 < \frac{π}{3} or \sin 1 < \sin \frac{π}{3} = 0.866 or \frac{1}{\sin 1} > 1.155 .$

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