The domain of `f (x) = log (|x - 2 | - 2 | - 1)` is

Options

A. R - (1,3)
B. `(- infty, -1) cup (1,3) cup (5, infty)`
C. `( 5, infty)`
D. None of these

Answer

To find the domain of the function \( f(x) = \log(|x - 2| - 2) - 1 \), we need to ensure that the argument of the logarithm is greater than zero. This means we need to solve the inequality:

\[ |x - 2| - 2 > 0 \]

### Step 1: Solve the inequality

First, we can rewrite the inequality:

\[ |x - 2| > 2 \]

### Step 2: Break it down into cases

The absolute value inequality \( |x - 2| > 2 \) can be split into two cases:

1. \( x - 2 > 2 \)
2. \( x - 2 < -2 \)

### Step 3: Solve each case

**Case 1:**

\[ x - 2 > 2 \implies x > 4 \]

**Case 2:**

\[ x - 2 < -2 \implies x < 0 \]

### Step 4: Combine the results

From the two cases, we find that:

- From Case 1, we have \( x > 4 \).
- From Case 2, we have \( x < 0 \).

Thus, the solution for the inequality \( |x - 2| > 2 \) is:

\[ x < 0 \quad \text{or} \quad x > 4 \]

### Step 5: Write the domain in interval notation

The domain of the function \( f(x) \) can be expressed in interval notation as:

\[ (-\infty, 0) \cup (4, \infty) \]

### Final Answer

The domain of \( f(x) = \log(|x - 2| - 2) - 1 \) is:

\[ (-\infty, 0) \cup (4, \infty) \]