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 will follow these steps:

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

**Hint:** Remember that the cosine function has a maximum value of 1 and a minimum value of -1.

### Step 2: Multiply the inequality by 2
Next, we multiply the entire inequality by 2:
\[ -2 \leq 2 \cos x \leq 2 \]

**Hint:** When multiplying an inequality by a positive number, the direction of the inequality remains the same.

### Step 3: Rearranging the inequality
Now, we rearrange the inequality by subtracting 1 from all parts:
\[ -2 - 1 \leq 2 \cos x - 1 \leq 2 - 1 \]
This simplifies to:
\[ -3 \leq 2 \cos x - 1 \leq 1 \]

**Hint:** Subtracting a constant from all parts of an inequality preserves the inequality's direction.

### Step 4: Change the signs and rearrange
Next, we can rearrange the inequality by multiplying by -1 (which reverses the inequality signs):
\[ 1 \geq - (2 \cos x - 1) \geq 3 \]
This can be rewritten as:
\[ 1 \geq 1 - 2 \cos x \geq 3 \]

**Hint:** Remember that multiplying or dividing by a negative number reverses the inequality.

### Step 5: Find the bounds for \( f(x) \)
Now we take the reciprocal of the inequality. When taking the reciprocal, we must reverse the inequalities:
\[ \frac{1}{3} \leq f(x) \leq 1 \]

**Hint:** When taking the reciprocal of a positive quantity, the direction of the inequality changes.

### Step 6: Determine the range of \( f(x) \)
From the above inequalities, we can conclude that the range of \( f(x) \) is:
\[ (-\infty, -1) \cup \left[\frac{1}{3}, \infty\right) \]

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