Find the range of the following functions:
`y=1/(2-cos3x)`

Answer

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

### Step 1: Determine the range of \( \cos(3x) \)
The cosine function oscillates between -1 and 1. Therefore, we have:
\[ -1 \leq \cos(3x) \leq 1 \]

### Step 2: Transform the inequality for \( 2 - \cos(3x) \)
Next, we will manipulate the inequality to find the range of \( 2 - \cos(3x) \):
\[ 2 - 1 \leq 2 - \cos(3x) \leq 2 - (-1) \]
This simplifies to:
\[ 1 \leq 2 - \cos(3x) \leq 3 \]

### Step 3: Take the reciprocal of the inequality
Now we will take the reciprocal of the entire inequality. Remember that when taking the reciprocal, the direction of the inequalities will change:
\[ \frac{1}{3} \leq \frac{1}{2 - \cos(3x)} \leq 1 \]
This means:
\[ \frac{1}{2 - \cos(3x)} \text{ will range from } \frac{1}{3} \text{ to } 1 \]

### Step 4: Relate back to \( y \)
Since \( y = \frac{1}{2 - \cos(3x)} \), we can conclude that:
\[ \frac{1}{3} \leq y \leq 1 \]

### Conclusion
Thus, the range of the function \( y = \frac{1}{2 - \cos(3x)} \) is:
\[ \boxed{\left[\frac{1}{3}, 1\right]} \]