The function takes an input string, doubles the second character, and replaces the first character with the second character.

[CODE]: 
```python
def fx_approximation(s: str) -> str:
    return s[1] + s[1] + s[2:]
```

Please note that there might be some differences between f(x) and this approximation, especially for special characters like in the "xy" case. The code above explains the main behavior observed in our tests.