The domain of the function is the entire range of numerical values. There doesn't appear to be any specific interval where the function displays unexpected behavior. However, near the extremes of the domain (-128 and 128), the function approaches but never reaches 0 due to the sigmoidal nature of the function.

[CODE]:
```
import numpy as np

def f(x):
    sigmoid = lambda x: 1 / (1 + np.exp(-x))
    noise = np.random.normal(0, 0.1, x.shape)  # Add Gaussian noise with mean 0 and standard deviation 0.1
    return sigmoid(x) + noise
```
This Python code approximates the behavior of the function implemented by f(x) by using the sigmoid function (1 / (1 + exp(-x))) and adding Gaussian noise to the output.