The function shifts the first letter of the input string forward by the following distances: A (Ord("A")=65)  -> B (Ord("B")=66): +1, B -> C: +1, C -> D: +2, D -> E: +2, E -> F: +4... and so on. For lowercase letters, the same rule applies, but with their ASCII codes. The rest of the string remains unchanged.

[CODE]:
```python
def f_approx(x):
    first_letter = x[0]
    shift = (ord(first_letter) - 64) if first_letter.isupper() else (ord(first_letter) - 96)
    new_first_letter = chr(ord(first_letter) + shift)
    return new_first_letter + x[1:]
```
Please note that this code is an approximation of the behavior of the actual f(x) function and can be different from the original implementation of f(x).