For a given string, if it starts with a lowercase letter, increment each character's ASCII value by 2. If it starts with an uppercase letter, decrement each character's ASCII value by 2. If it starts with a digit, increment the first digit by 1. If it starts with a special character, swap the first two characters.

[CODE]:
```python
def approx_function(s: str) -> str:
    if s[0].islower():
        return ''.join([chr(ord(x) + 2) for x in s])
    elif s[0].isupper():
        return ''.join([chr(ord(x) - 2) for x in s])
    elif s[0].isdigit():
        return str(int(s[0]) + 1) + s[1:]
    else:
        return s[1] + s[0] + s[2:]
```

Keep in mind that this code is an approximation based on the given input-output examples and may not cover all possible edge cases.