The function sorts the first half of the input string in ascending alphabetical order while keeping the second half of the input string the same, and duplicates the last character of the input string.

[CODE]:
```python
def approximate_function(s: str) -> str:
    first_half = sorted(s[:len(s) // 2])
    second_half = s[len(s) // 2:]
    return ''.join(first_half) + second_half + s[-1]
```