Draw the graphs of the following function
`f(x)=abs(lnabsx)x inR-{0}`

Answer

To draw the graph of the function \( f(x) = |\ln |x| | \) for \( x \in \mathbb{R} \setminus \{0\} \), we will follow these steps:

### Step 1: Understand the function
The function \( f(x) = |\ln |x| | \) involves the natural logarithm of the absolute value of \( x \). The absolute value ensures that the function is defined for both positive and negative values of \( x \), excluding zero.

### Step 2: Analyze \( \ln |x| \)
1. For \( x > 0 \):
- The function simplifies to \( f(x) = \ln x \).
- The graph of \( \ln x \) is defined for \( x > 0 \) and has the following characteristics:
- It approaches \( -\infty \) as \( x \) approaches \( 0 \).
- It crosses the x-axis at \( x = 1 \) (since \( \ln(1) = 0 \)).
- It increases without bound as \( x \) increases.

2. For \( x < 0 \):
- The function simplifies to \( f(x) = \ln (-x) \) (since \( |x| = -x \) for \( x < 0 \)).
- The graph of \( \ln (-x) \) is defined for \( x < 0 \) and has similar characteristics to \( \ln x \):
- It approaches \( -\infty \) as \( x \) approaches \( 0 \) from the left.
- It crosses the x-axis at \( x = -1 \) (since \( \ln(1) = 0 \)).
- It increases without bound as \( x \) decreases.

### Step 3: Apply the absolute value
Since we are taking the absolute value of \( \ln |x| \):
- For \( x > 1 \), \( f(x) = \ln x \) (positive values).
- For \( 0 < x < 1 \), \( f(x) = -\ln x \) (since \( \ln x \) is negative).
- For \( x < -1 \), \( f(x) = \ln (-x) \) (positive values).
- For \( -1 < x < 0 \), \( f(x) = -\ln (-x) \) (since \( \ln (-x) \) is negative).

### Step 4: Sketch the graph
1. **For \( x > 1 \)**:
- The graph of \( f(x) = \ln x \) is in the first quadrant and increases.

2. **For \( 0 < x < 1 \)**:
- The graph of \( f(x) = -\ln x \) is in the first quadrant and decreases from \( \infty \) to \( 0 \) as \( x \) approaches \( 1 \).

3. **For \( x < -1 \)**:
- The graph of \( f(x) = \ln (-x) \) is in the second quadrant and increases.

4. **For \( -1 < x < 0 \)**:
- The graph of \( f(x) = -\ln (-x) \) is in the second quadrant and decreases from \( \infty \) to \( 0 \) as \( x \) approaches \( -1 \).

### Final Graph
The final graph will be symmetric about the y-axis, with the following key points:
- At \( x = 1 \) and \( x = -1 \), \( f(x) = 0 \).
- As \( x \) approaches \( 0 \) from either side, \( f(x) \) approaches \( \infty \).