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 follow these steps:

### Step 1: Determine the Domain
The function involves a logarithm, \( \ln(|x| - 1) \). The logarithm is defined only for positive arguments. Therefore, we need to find when \( |x| - 1 > 0 \).

1. **Set up the inequality**:
\[ |x| - 1 > 0 \]
This simplifies to:
\[ |x| > 1 \]

2. **Solve the absolute value inequality**:
This means:
\[ x < -1 \quad \text{or} \quad x > 1 \]
Therefore, the domain of \( f(x) \) is:
\[ (-\infty, -1) \cup (1, \infty) \]

### Step 2: Determine the Range
Next, we analyze the range of the function \( f(x) \).

1. **Examine the logarithm**:
The expression \( \ln(|x| - 1) \) can take any real value from \( -\infty \) to \( \infty \) as \( |x| \) approaches \( 1 \) from the right (for \( x > 1 \) or \( x < -1 \)).

2. **Apply the modulus**:
Since we have \( |\ln(|x| - 1)| \), the output of this expression will always be non-negative:
\[ |\ln(|x| - 1)| \geq 0 \]
Therefore, the values of \( f(x) \) can be expressed as:
\[ f(x) = 1 - |\ln(|x| - 1)| \]
As \( |\ln(|x| - 1)| \) approaches \( 0 \) (when \( |x| \) is just above \( 1 \)), \( f(x) \) approaches \( 1 \). As \( |\ln(|x| - 1)| \) approaches \( \infty \) (as \( |x| \) increases), \( f(x) \) approaches \( -\infty \).

3. **Determine the range**:
Thus, the range of \( f(x) \) is:
\[ (-\infty, 1] \]

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