The function reverses the input string and removes some special characters (excluding numbers, periods, and a few select characters like !, @, *, and ^).

[CODE]:
```python
def approx_function(input_string):
    allowed_chars = 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789.!*^'
    reversed_string = input_string[::-1]
    result = "".join(char for char in reversed_string if char in allowed_chars)
    return result
```