The function takes a string input and returns a new string with each alphabetic character modified by adding a constant value, while keeping non-alphabetic characters unchanged. The function is case-insensitive.

[CODE]:
```python
def approximate_function(input_string):
    output = ''
    constant_shift = 9  # This value of 9 was derived from the examples provided (e.g., a -> j, A -> j)
    for char in input_string:
        if char.isalpha():
            new_char = chr(((ord(char.lower()) - ord('a') + constant_shift) % 26) + ord('a'))
            output += new_char
        else:
            output += char
    return output
```

Please confirm if this description and code approximation match the behavior of the function in the ./temp/function.py script.