Based on the given input-output pairs, I will describe the function's behavior and provide a Python code approximation.

[DESCRIPTION]:
The function takes a string as input and does the following:
1. Swaps the first character with the last character.
2. If the new last character is a digit, increment it by one. If the digit is 9, replace it with 0.
3. If the new last character is a lowercase letter and not 'z', increment it by one.
4. If the new last character is 'z', replace it with 'a'.

[CODE]:
```python
def approximate_function(s):
    if not s:
        return s

    s = list(s)
    s[0], s[-1] = s[-1], s[0]

    if s[-1].isdigit():
        s[-1] = str((int(s[-1]) + 1) % 10)
    elif s[-1] == 'z':
        s[-1] = 'a'
    elif s[-1].islower():
        s[-1] = chr(ord(s[-1]) + 1)

    return ''.join(s)
```

Note that this code is an approximation of the function's behavior based on the provided input-output pairs.