The function doubles the second and third-to-last characters in the input string and appends a "q" at the end, leaving the rest of the characters unchanged. The function does not process input strings containing punctuation characters.

[CODE]: 
```python
def f_approx(x: str):
    if any(char in x for char in "!?.,:"):
        return "Error: Undefined input"
    else:
        x = list(x)
        if len(x) > 1:
            x.insert(1, x[1])
        if len(x) > 2:
            x.insert(-2, x[-3])
        x.append("q")
        return "".join(x)
```

Now I will perform further testing of my approximation by running PYTHON(./temp/function.py "input1" "input2") examples.

COMMAND: PYTHON(./temp/function.py "abcde" "uvwxy" "mix3s" "cap1tal" "7let9ters")