Find the range of the following functions: (where {.} and [.] represent fractional part and greatest integer part functions respectively)
`f(x)=1/(2-cos 3x)`

Explanation

To find the range of the function \( f(x) = \frac{1}{2 - \cos(3x)} \), we will follow these steps:

### Step 1: Analyze the Denominator
The first step is to analyze the expression in the denominator, which is \( 2 - \cos(3x) \). We know that the cosine function oscillates between -1 and 1. Therefore, we can find the minimum and maximum values of \( 2 - \cos(3x) \).

### Step 2: Determine the Range of the Denominator
Since \( \cos(3x) \) ranges from -1 to 1, we can substitute these values into \( 2 - \cos(3x) \):

- When \( \cos(3x) = 1 \):
\[ 2 - \cos(3x) = 2 - 1 = 1 \]

- When \( \cos(3x) = -1 \):
\[ 2 - \cos(3x) = 2 - (-1) = 3 \]

Thus, the range of \( 2 - \cos(3x) \) is from 1 to 3:
\[ 1 \leq 2 - \cos(3x) \leq 3 \]

### Step 3: Find the Range of the Function
Next, we need to find the range of the function \( f(x) = \frac{1}{2 - \cos(3x)} \). The function \( f(x) \) will attain its maximum value when the denominator \( 2 - \cos(3x) \) is at its minimum, and it will attain its minimum value when the denominator is at its maximum.

- **Maximum of \( f(x) \)** occurs when \( 2 - \cos(3x) \) is minimum (which is 1):
\[ f(x)_{\text{max}} = \frac{1}{1} = 1 \]

- **Minimum of \( f(x) \)** occurs when \( 2 - \cos(3x) \) is maximum (which is 3):
\[ f(x)_{\text{min}} = \frac{1}{3} \]

### Step 4: Conclusion
Therefore, the range of the function \( f(x) = \frac{1}{2 - \cos(3x)} \) is:
\[ \left[\frac{1}{3}, 1\right] \]