The function capitalizes the first character of the input string and, for all uppercase alphabetical characters in the original string, swaps them with the previous character in the modified string.

[CODE]: 
```
def approximate_function(s: str) -> str:
    if not s:
        return s
    modified_string = list(s.capitalize())
    for i in range(1, len(modified_string)):
        if s[i-1].isupper():
            modified_string[i-1], modified_string[i] = modified_string[i], modified_string[i-1]
    return ''.join(modified_string)
```