The function can accept any numerical input within the range of [-128, 128]. However, there is an interesting behavior observed around the interval [-32, 32]. The function displays larger fluctuations and more significant changes in the output within this interval.

[CODE]:

```python
def f(x):
    if x >= -32 and x <= 32:
        return x**3 + 3*x**2 + 2*x + 3.7
    else:
        return 1.5 * (x ** 2) - 2 * x
```

Please note that this is just an approximation based on the observed behavior and may not perfectly represent the actual function implemented in the script.