The function takes an input string, appends "a" to the input string, doubles the first character, and adds 2 to the ASCII value of the first character.

[CODE]:
```python
def f_approx(x):
    first_char_ascii = ord(x[0])
    doubled_first_char = chr(first_char_ascii + 2)
    return doubled_first_char + x[1:] + "a"
```

Please test the function f_approx on your input strings to see if it behaves similarly to the observed output of f(x).