The function seems to increment each alphabetical character in the input string by 2 positions in the alphabet, wrapping around when it reaches the end of the alphabet (for both upper and lowercase letters). It leaves numbers and symbols unchanged. If the input string contains only numbers or is empty, the output is an empty string.

[CODE]:
```python
def approximate_function(input_string):
    output_string = ""
    for char in input_string:
        if char.isalpha():
            shift = 2
            if char.isupper():
                output_string += chr((ord(char) - ord('A') + shift) % 26 + ord('A'))
            else:
                output_string += chr((ord(char) - ord('a') + shift) % 26 + ord('a'))
        else:
            output_string += char
    return output_string
```