The function f(x) shifts each lowercase alphabetic character in the input string back by 4 positions in the alphabet (with wrap-around) while shifting each uppercase alphabetic character ahead by 2 positions in the alphabet (with wrap-around). Non-alphabetic characters remain unchanged.

[CODE]:
```python
def f(x):
    result = ''
    for char in x:
        if 'a' <= char <= 'z':
            result += chr(((ord(char) - ord('a') - 4) % 26) + ord('a'))
        elif 'A' <= char <= 'Z':
            result += chr(((ord(char) - ord('A') + 2) % 26) + ord('A'))
        else:
            result += char
    return result
```