The function appears to increment the first, third, fifth characters (and so on) of the input string by 2, and leave the others unchanged. The operation wraps around for lowercase, uppercase and special characters above space (32 in ASCII).

[CODE]:
```python
def approximate_fx(input_str):
    output_str = ""
    for i, c in enumerate(input_str):
        if i % 2 == 0:
            if 'a' <= c <= 'z':
                new_c = chr(((ord(c) - ord('a') + 2) % 26) + ord('a'))
            elif 'A' <= c <= 'Z':
                new_c = chr(((ord(c) - ord('A') + 2) % 26) + ord('A'))
            elif ' ' < c < '\x7f':
                new_c = chr(((ord(c) - 33 + 2) % 94) + 33)
            else:
                new_c = c
        else:
            new_c = c
        output_str += new_c
    return output_str
```

This code attempts to approximate the behavior of the function f(x).