The function increments each lower case letter in the given input by 4 positions, with wraparound after 'z', while leaving all other characters unchanged.

[CODE]: 
```python
def approximate_function(input_string):
    output = ''
    for char in input_string:
        if char.islower():
            shifted_char = chr(((ord(char) - ord('a') + 4) % 26) + ord('a'))
            output += shifted_char
        else:
            output += char
    return output
```

To confirm our understanding, let's run one more test with a more diverse input.

COMMAND: PYTHON(./temp/function.py aZ1!zZ)