The function shifts each alphabetic character in the input string forward by 3 positions if it's a lowercase letter, and backward by 4 positions if it's an uppercase letter. Numbers and special characters remain unchanged.

[CODE]: 
```python
def approximate_function(input_str):
    output = ''
    for ch in input_str:
        if ch.islower():
            output += chr(((ord(ch) - ord('a') + 3) % 26) + ord('a'))
        elif ch.isupper():
            output += chr(((ord(ch) - ord('A') - 4) % 26) + ord('A'))
        else:
            output += ch
    return output
```