The function takes a string input and appends the next character in the ASCII sequence after the first character of the input string. If the length of the input string is odd, it also appends the second character of the input string.

[CODE]:
```python
def approximate_function(s: str) -> str:
    first_char_shifted = chr(ord(s[0]) + 1)
    result = s + first_char_shifted
    if len(s) % 2 == 1:
        result += s[1]
    return result
```