The function shifts lowercase letters by 1 position in the alphabet, converts uppercase letters to lowercase, keeps all other characters unchanged, and appends "ynZ" to the end.

[CODE]:
```python
def approx_function(s: str) -> str:
    result = ""
    for c in s:
        if 'a' <= c <= 'z':
            shifted = chr(((ord(c) - ord('a') + 1) % 26) + ord('a'))
            result += shifted
        elif 'A' <= c <= 'Z':
            result += c.lower()
        else:
            result += c
    result += "ynZ"
    return result
```

Let's test the behavior of the approximated function on some more inputs.

COMMAND: PYTHON(./temp/function.py "testFunction" "shifting" "UpperCase" "mixedCase123!" "specialChars()")