The function takes an input string and returns a new string where each letter is shifted by a fixed position in the alphabet (preserving the case), and non-letter characters are left unchanged.

[CODE]:
```python
def approximate_function(input_str):
    alphabet = 'abcdefghijklmnopqrstuvwxyz'
    shift = 9
    output = ''

    for char in input_str:
        if char.isalpha():
            if char.islower():
                output += alphabet[(alphabet.index(char) + shift) % 26]
            else:
                upper_alphabet = alphabet.upper()
                output += upper_alphabet[(upper_alphabet.index(char) + shift) % 26]
        else:
            output += char

    return output
```

Please perform more tests if you wish, or let me know if there's anything else I can help with.