Range of the function f defined by
`(x) =[(1)/(sin{x})]` (where `[.]` and `{x}` denotes
greatest integer and fractional part of x respectively) is

Options

A. I,the set of integers
B. N,the set of natural number
C. W, the set of whole number
D. `{2,3,4,…..}`

Explanation

To find the range of the function \( f(x) = \left\lfloor \frac{1}{\sin \{x\}} \right\rfloor \), where \( \{x\} \) denotes the fractional part of \( x \), we can follow these steps:

### Step 1: Understand the fractional part of \( x \)
The fractional part of \( x \), denoted as \( \{x\} \), is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
This means \( \{x\} \) takes values in the interval \( [0, 1) \).

### Step 2: Analyze the sine function
The sine function, \( \sin \{x\} \), will take values in the interval \( [0, \sin(1)] \) for \( \{x\} \) in \( [0, 1) \). The sine function is continuous and increasing in this interval, reaching its maximum at \( \sin(1) \).

### Step 3: Determine the range of \( \frac{1}{\sin \{x\}} \)
Since \( \sin \{x\} \) approaches \( 0 \) as \( \{x\} \) approaches \( 0 \), \( \frac{1}{\sin \{x\}} \) will approach \( +\infty \). The minimum value occurs when \( \{x\} \) is at its maximum, which is \( \sin(1) \). Therefore, the minimum value of \( \frac{1}{\sin \{x\}} \) is:
\[ \frac{1}{\sin(1)} \]

### Step 4: Analyze the greatest integer function
The function \( f(x) = \left\lfloor \frac{1}{\sin \{x\}} \right\rfloor \) will take integer values. Since \( \frac{1}{\sin \{x\}} \) can take any value from \( \frac{1}{\sin(1)} \) to \( +\infty \), we need to find the smallest integer greater than or equal to \( \frac{1}{\sin(1)} \).

### Step 5: Conclusion on the range
The range of \( f(x) \) will be all natural numbers \( n \) such that:
\[ n \geq \left\lceil \frac{1}{\sin(1)} \right\rceil \]
Since \( \sin(1) \) is a positive number less than \( 1 \), \( \frac{1}{\sin(1)} \) is greater than \( 1 \). Thus, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = \{ n \in \mathbb{N} : n \geq \left\lceil \frac{1}{\sin(1)} \right\rceil \} \]