Range of `f(x)=(1)/(1-2 cos x) is :`

Options

A. `[1/3,1]`
B. `[-1,1/3]`
C. `(-oo, 1] cup [1/3, oo)`
D. `[-1/3,1]`

Answer

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

### Step 1: Determine the range of \( \cos x \)
The cosine function oscillates between -1 and 1 for all real \( x \):
\[ -1 \leq \cos x \leq 1 \]

### Step 2: Substitute the range of \( \cos x \) into the function
We substitute the values of \( \cos x \) into the expression \( 1 - 2 \cos x \):
- When \( \cos x = 1 \):
\[ 1 - 2 \cdot 1 = 1 - 2 = -1 \]
- When \( \cos x = -1 \):
\[ 1 - 2 \cdot (-1) = 1 + 2 = 3 \]

Thus, we have:
\[ -1 \leq 1 - 2 \cos x \leq 3 \]

### Step 3: Analyze the expression \( 1 - 2 \cos x \)
From the above step, we can conclude:
\[ 1 - 2 \cos x \text{ ranges from } -1 \text{ to } 3 \]

### Step 4: Invert the range to find the range of \( f(x) \)
Since \( f(x) = \frac{1}{1 - 2 \cos x} \), we need to consider the behavior of the function as \( 1 - 2 \cos x \) approaches its bounds:
- As \( 1 - 2 \cos x \) approaches -1 (from the right), \( f(x) \to \frac{1}{-1} = -1 \).
- As \( 1 - 2 \cos x \) approaches 3 (from the left), \( f(x) \to \frac{1}{3} \).

### Step 5: Determine the limits and intervals
- When \( 1 - 2 \cos x \) is just above -1, \( f(x) \) approaches -1 but never reaches it.
- When \( 1 - 2 \cos x \) is just below 3, \( f(x) \) approaches \( \frac{1}{3} \) but never reaches it.

Thus, we can conclude:
\[ f(x) \text{ can take values in the intervals } (-\infty, -1) \cup \left(\frac{1}{3}, \infty\right) \]

### Final Answer
The range of \( f(x) = \frac{1}{1 - 2 \cos x} \) is:
\[ (-\infty, -1) \cup \left(\frac{1}{3}, \infty\right) \]