The function takes a string input and applies a Caesar cipher with a varying shift amount to alphabetical characters. The shift amount starts at 0, increases by 5 for each alphabetical position, and resets after reaching a maximum of 25. Non-alphabetical characters remain untouched.

[CODE]:
```python
def approximate_function(s: str) -> str:
    output = []
    for c in s:
        if c.isalpha():
            shift = (ord(c.lower()) - ord('a') + 5) % 26
            shifted_char = chr(ord(c) + shift)
            if c.isupper() and shifted_char.islower() or c.islower() and shifted_char.isupper():
                shifted_char = chr(ord(shifted_char) - 26 * (shift // abs(shift)))
            output.append(shifted_char)
        else:
            output.append(c)
    return ''.join(output)
```

Please let me know if you would like further testing or analysis.