To describe the function implemented by f(x), we can run experiments on it. We will call the function on multiple inputs within the range [-128, 128] and analyze the outputs.

[DESCRIPTION]:
Based on the experiments, the function appears to be implementing a sigmoid-like behavior. The output starts close to -1 for negative inputs, approaches 0 around x=0, and then approaches 1 for positive inputs. However, there is an interval where the function displays unexpected behavior. For values of x in the range [-4, 4], the function seems to oscillate rapidly between -1 and 1.

[DOMAIN]:
The domain of the function is the set of all real numbers, as there are no constraints given on the input. However, within the domain, there is an interval of interest [-4, 4] where the function behaves unexpectedly.

[CODE]:
An approximation of the behavior of the function can be described using a sigmoid function. Here's the Python code for the approximation:

```python
import math

def approx_function(x):
    return (2 / (1 + math.exp(-x))) - 1

```
This approximation uses the sigmoid function formula to transform the input x to the range [-1, 1]. The sigmoid function is defined as f(x) = 2 / (1 + e^(-x)) - 1.