The function takes a string input and increments its alphabetical characters by an offset of 8, wrapping around if needed, while leaving numeric characters unchanged.

[CODE]: 
```python
def approximate_function(input_string):
    output = ''
    for char in input_string:
        if char.isalpha():
            if char.isupper():
                output += chr(((ord(char) - ord('A') + 8) % 26) + ord('A'))
            else:
                output += chr(((ord(char) - ord('a') + 8) % 26) + ord('a'))
        elif char.isnumeric():
            output += char
    return output
```

Now, let's run some additional tests to further confirm our findings:

COMMAND: PYTHON(./temp/function.py "D" "G" "T" "z" "p" "5")