The function takes a string as input and returns a transformed version of the string. For strings containing only alphabetic characters, each character is shifted in the alphabet. For strings containing alphanumeric characters, the output is the same as the input. For strings representing boolean values, the output is a random string of alphabetic characters.

[CODE]:
```python
def f(x):
    if x.isalpha():
        output = ""
        for char in x:
            shifted_char = chr(ord(char) + 2)
            output += shifted_char
        return output
    elif x.isalnum():
        return x
    elif x.lower() == "true" or x.lower() == "false":
        return random_string()
```

Note: In the code, `random_string()` refers to a function that generates a random string of alphabetic characters.