The domain of the function seems to be all real numbers, as the function produces output values for any input value given. However, there is an interval where the function behaves differently, which is centered around zero. For the interval (-4, 4), the function displays unexpected behavior, deviating from the general pattern observed in the rest of the domain. Within this interval, the output values are clustered closely together and approach a constant value (-39.0).

[CODE]:

```
def f(x):
    if x >= -4 and x <= 4:
        return -39.0
    else:
        return -abs(x) - (1/8)*x**3
```

This code snippet approximates the behavior of the function f(x) by handling the special case for the interval (-4, 4) separately and using the general pattern observed for the rest of the domain. The function returns -39.0 within the interval and follows a non-linear decay trend for input values outside the interval.