The function seems to perform a character-wise Caesar cipher shift on alphabetical characters, while it leaves numeric and special characters unchanged. Each alphabetical character is shifted by 9 positions forward in the alphabet.

[CODE]: 
```python
def approximate_function(input_string):
    output_string = []
    for char in input_string:
        if char.isalpha():
            shift = 9
            if char.islower():
                output_string.append(chr(((ord(char) - ord('a') + shift) % 26) + ord('a')))
            else:
                output_string.append(chr(((ord(char) - ord('A') + shift) % 26) + ord('A')))
        else:
            output_string.append(char)
    return ''.join(output_string)
```