The function applies a Caesar cipher with a fixed shift to the alphabetic characters of an input string, preserving the original case, and leaves non-alphabetic characters unchanged.

[CODE]:
```python
def approximate_function(input_str):
    shift = 17  # Based on the observed outputs
    output_str = ''
    for char in input_str:
        if char.isalpha():
            ascii_offset = 65 if char.isupper() else 97
            output_str += chr(((ord(char) - ascii_offset + shift) % 26) + ascii_offset)
        else:
            output_str += char
    return output_str
```