Find the range of each of the following functions: (where {.} and [.] represents fractional part and greatest integer part functions respectively)
`f(x)=[1/(sin{x})]`

Explanation

To find the range of the function \( f(x) = \left[ \frac{1}{\sin(\{x\})} \right] \), where \( \{x\} \) represents the fractional part of \( x \), we will follow these steps:

### Step 1: Understand the fractional part function
The fractional part function \( \{x\} \) is defined as:
\[ \{x\} = x - [x] \]
This means that \( \{x\} \) always lies in the interval \([0, 1)\).

### Step 2: Determine the range of \( \sin(\{x\}) \)
Since \( \{x\} \) is in the interval \([0, 1)\), we need to evaluate \( \sin(\{x\}) \) over this interval. The sine function is continuous and increasing in this range:
- At \( \{x\} = 0 \), \( \sin(0) = 0 \).
- At \( \{x\} = 1 \), \( \sin(1) \) is approximately \( 0.8415 \).

Thus, the range of \( \sin(\{x\}) \) is:
\[ \sin(\{x\}) \in (0, \sin(1)) \approx (0, 0.8415) \]

### Step 3: Analyze \( \frac{1}{\sin(\{x\})} \)
Now, we need to find the range of \( \frac{1}{\sin(\{x\})} \). Since \( \sin(\{x\}) \) is positive and approaches \( 0 \) as \( \{x\} \) approaches \( 0 \), \( \frac{1}{\sin(\{x\})} \) will approach infinity. Conversely, when \( \sin(\{x\}) \) approaches \( \sin(1) \), \( \frac{1}{\sin(\{x\})} \) approaches \( \frac{1}{\sin(1)} \).

Thus, the range of \( \frac{1}{\sin(\{x\})} \) is:
\[ \frac{1}{\sin(\{x\})} \in \left( \frac{1}{\sin(1)}, \infty \right) \approx (1.188, \infty) \]

### Step 4: Apply the greatest integer function
Now we apply the greatest integer function \( [\cdot] \) to \( \frac{1}{\sin(\{x\})} \). The greatest integer function takes any real number and gives the largest integer less than or equal to that number.

Since \( \frac{1}{\sin(\{x\})} \) ranges from approximately \( 1.188 \) to infinity, the smallest integer greater than \( 1.188 \) is \( 2 \). Therefore, the range of \( [\frac{1}{\sin(\{x\})}] \) will include all integers starting from \( 2 \) to infinity.

### Final Answer
Thus, the range of the function \( f(x) = \left[ \frac{1}{\sin(\{x\})} \right] \) is:
\[ \text{Range of } f(x) = \{2, 3, 4, \ldots\} = \mathbb{N} \setminus \{1\} \]