The function f(x) shifts each character of the input string by a fixed number of positions in the alphabet.

[CODE]:
```python
def f(x):
    result = ""
    shift = 1  # The fixed number of positions to shift each character
    
    for char in x:
        if char.isalpha():
            new_char = chr((ord(char) - ord('a') + shift) % 26 + ord('a'))  # Shift the character in the alphabet
            result += new_char
        else:
            result += char
    
    return result
```

Please note that this code assumes the input string consists of lowercase alphabetical characters. It skips any non-alphabetical characters without any transformation. The shift value used in the code above is arbitrarily set to 1, but it might be different in the actual implementation of the function.