The function replaces the first character of the input string with the character that is its ASCII code minus 1 if it's a lowercase letter or with the corresponding uppercase letter 25 positions after it if it's an uppercase letter.

[CODE]: 
```python
def approximate_function(input_string):
    first_char = input_string[0]

    if first_char.isalpha():
        if first_char.islower():
            new_char = chr(ord(first_char)-1)
        else:
            new_char = chr(((ord(first_char) - ord('A') + 25) % 26) + ord('A'))
    else:
        new_char = first_char

    return new_char + input_string[1:]
```

Please let me know if any additional tests are needed to confirm the behavior.