The function f(x) takes an input string and returns a new string where each character is shifted to another character by a constant number of positions in the alphabet. It is case-insensitive and does not affect numbers or special characters.

[CODE]: 
```python
def approximate_function(input_string):
    shift = 3
    output = ""

    for char in input_string:
        if char.isalpha():
            shifted_char = chr(((ord(char.lower()) - ord('a') + shift) % 26) + ord('a'))
            output += shifted_char if char.islower() else shifted_char.upper()
        else:
            output += char

    return output
```

Please note that this code is an approximation of the behavior and might not exactly represent the original function.