The function takes an input string and applies the following transformations:
- Shift each alphabetic character by 2 positions in the alphabet (wrapping around if needed).
- Keep non-alphabetic characters unchanged.
- Insert the letter 'Y' after each shifted character.

[CODE]:
```python
def transform(character):
    if 'a' <= character <= 'z':
        return chr(((ord(character) - ord('a') + 2) % 26) + ord('a'))
    elif 'A' <= character <= 'Z':
        return chr(((ord(character) - ord('A') + 2) % 26) + ord('A'))
    else:
        return character


def approx_function(input_string):
    transformed_string = ''
    for character in input_string:
        transformed_string += transform(character) + 'Y'
    
    return transformed_string[:-1]  # Remove the last 'Y' before returning the result
```

This code should approximate the behavior of f(x) as seen in the responses.