The function takes the input string and increments each character by 6 positions in the alphabet. If the character is not a letter, it remains unchanged. If the alphabet incrementation goes beyond 'z', it loops back to 'a'.

[CODE]: 
def approx_function(input_string):
    output_string = ""
    for char in input_string:
        if char.isalpha():
            increment = 6
            if char.isupper():
                output_string += chr(((ord(char) - ord('A') + increment) % 26) + ord('A'))
            else:
                output_string += chr(((ord(char) - ord('a') + increment) % 26) + ord('a'))
        else:
            output_string += char
    return output_string