The function `f(x)` takes an input string and applies a set of transformations to it. For lowercase alphabetic characters, it appears to substitute each character with another character that is a fixed distance away in the alphabet. In particular, it looks like the character's ASCII value is increased by replacing it with the n-th next character, where n is a fixed number. Uppercase characters are first converted to lowercase and then processed in the same manner. For numerical characters and special characters, there are a variety of transformations such as reversing the order or swapping characters. White spaces are preserved as is.

[CODE]:
```python
def f_approx(x):
    output = []
    for c in x:
        if c.isalpha():
            lowercase_c = c.lower()
            new_c = chr(((ord(lowercase_c) - ord('a') + 2) % 26) + ord('a'))
            output.append(new_c)
        elif c.isdigit():
            output.insert(0, c)
        else:
            i = output.index(c) if c in output else len(output)
            output.insert(i, c)
    return ''.join(output)
```