What is the domain and range of the function `f(x)=1-|ln(|x|-1)|`

Answer

To find the domain and range of the function \( f(x) = 1 - |\ln(|x| - 1)| \), we will analyze the function step by step.

### Step 1: Determine the domain of \( f(x) \)

1. **Identify the logarithmic function**: The expression inside the logarithm is \( |x| - 1 \). For the logarithm to be defined, we need:
\[ |x| - 1 > 0 \implies |x| > 1 \]
This means \( x < -1 \) or \( x > 1 \).

2. **Combine the intervals**: Therefore, the domain of \( f(x) \) is:
\[ (-\infty, -1) \cup (1, \infty) \]

### Step 2: Determine the range of \( f(x) \)

1. **Analyze the expression \( |\ln(|x| - 1)| \)**:
- As \( |x| \) approaches 1 from the right (i.e., \( x \to 1^+ \) or \( x \to -1^- \)), \( |x| - 1 \) approaches 0, and \( \ln(|x| - 1) \) approaches \( -\infty \). Therefore, \( |\ln(|x| - 1)| \) approaches \( +\infty \).
- As \( |x| \) increases (i.e., \( x \to \infty \) or \( x \to -\infty \)), \( |x| - 1 \) becomes large, and \( \ln(|x| - 1) \) approaches \( +\infty \). Thus, \( |\ln(|x| - 1)| \) also approaches \( +\infty \).

2. **Evaluate \( f(x) \)**:
- Since \( f(x) = 1 - |\ln(|x| - 1)| \), as \( |\ln(|x| - 1)| \) approaches \( +\infty \), \( f(x) \) approaches \( -\infty \).
- When \( |x| \) is just greater than 1 (for example, \( x = 2 \)), we can calculate:
\[ f(2) = 1 - |\ln(2 - 1)| = 1 - |\ln(1)| = 1 - 0 = 1 \]
- Thus, the maximum value of \( f(x) \) occurs at \( x = 2 \) or \( x = -2 \) and is equal to 1.

3. **Combine the findings**: Since \( f(x) \) can take values from \( -\infty \) up to 1, the range of \( f(x) \) is:
\[ (-\infty, 1] \]

### Final Result

- **Domain**: \( (-\infty, -1) \cup (1, \infty) \)
- **Range**: \( (-\infty, 1] \)