The domain of the function is the entire real number line, except for x = 0. The function displays unexpected behavior at x = 0, where it becomes infinite.

[CODE]:

```python
import math

def f(x):
    if x == 0:
        return float('inf')
    elif x < 0:
        return float('nan')
    else:
        return -math.log(x)
```

This code approximates the behavior of the function. The code checks if x is equal to 0 and returns infinity in that case. If x is less than 0, it returns NaN. Otherwise, it computes the natural logarithm of x and returns its negative value.