To find the range of the function \( f(x) = \frac{1}{1 - 2 \cos x} \), we will follow these steps:
### Step 1: Determine the range of \( \cos x \)
The cosine function has a known range:
\[
-1 \leq \cos x \leq 1
\]
### Step 2: Transform the range of \( \cos x \)
Now, we will transform the range of \( \cos x \) by multiplying by -2:
\[
-2 \leq -2 \cos x \leq 2
\]
Reversing the inequalities (since we multiplied by a negative number):
\[
2 \geq -2 \cos x \geq -2
\]
### Step 3: Add 1 to the entire inequality
Next, we add 1 to all parts of the inequality:
\[
2 + 1 \geq -2 \cos x + 1 \geq -2 + 1
\]
This simplifies to:
\[
3 \geq 1 - 2 \cos x \geq -1
\]
### Step 4: Take the reciprocal
Now, we take the reciprocal of the entire inequality. Since \( 1 - 2 \cos x \) can take values between -1 and 3, we need to consider the signs:
- When \( 1 - 2 \cos x \) is positive, the inequality sign remains the same.
- When \( 1 - 2 \cos x \) is negative, the inequality sign reverses.
From the previous step, we see that \( 1 - 2 \cos x \) can be zero when \( \cos x = \frac{1}{2} \) (which occurs at \( x = \frac{\pi}{3} + 2k\pi \) and \( x = -\frac{\pi}{3} + 2k\pi \) for integers \( k \)). Thus, we need to consider the intervals where \( 1 - 2 \cos x \) is positive and negative.
### Step 5: Analyze the intervals
1. When \( 1 - 2 \cos x > 0 \):
- This occurs when \( \cos x < \frac{1}{2} \) (i.e., \( x \) is in the intervals \( \left(\frac{\pi}{3}, \frac{5\pi}{3}\right) \)).
- In this case, \( 1 - 2 \cos x \) ranges from \( 0 \) to \( 3 \).
2. When \( 1 - 2 \cos x < 0 \):
- This occurs when \( \cos x > \frac{1}{2} \) (i.e., \( x \) is in the intervals \( \left(-\frac{\pi}{3}, \frac{\pi}{3}\right) \) and \( \left(\frac{5\pi}{3}, \frac{7\pi}{3}\right) \)).
- In this case, \( 1 - 2 \cos x \) ranges from \( -1 \) to \( 0 \).
### Step 6: Combine the results
Taking the reciprocal of the positive range \( (0, 3) \):
\[
\frac{1}{3} < f(x) < \infty
\]
Taking the reciprocal of the negative range \( (-1, 0) \):
\[
-\infty < f(x) < -1
\]
### Final Result
Combining both ranges, we find that the range of \( f(x) \) is:
\[
(-\infty, -1) \cup \left(\frac{1}{3}, \infty\right)
\]
