The function f(x) takes a string input, removes any digits or special characters, and replaces the remaining letters with lowercase letters shifted by a certain number of positions in the alphabet.

Now, let's implement an approximation of the behavior observed in the function.

[CODE]:
```python
def approximate_fx(input_string):
    result = ""
    for char in input_string:
        if char.isalpha():
            new_char = chr(((ord(char.lower()) - ord('a') + 2) % 26) + ord('a'))
            result += new_char
        else:
            continue
    return result
```

Please note that this approximation might not be exact, but it should work for the majority of the inputs provided and demonstrate the observed behavior.