For any input string, the function returns 'zeroz' if the string contains only numbers, and if the string contains any non-numeric character, it returns a string corresponding to the first non-numeric character ('a' -> 'oneo', 'b' -> 'twot', 'c' -> 'threet').

[CODE]:
```python
def approximate_function(input_str):
    if input_str.isnumeric():
        return 'zeroz'
    else:
        char_to_string = {'a': 'oneo', 'b': 'twot', 'c': 'threet'}
        for char in input_str:
            if char in char_to_string:
                return char_to_string[char]
    return None
```

I'd like to test a few more cases to ensure the current understanding is correct, especially for mixed cases.

COMMAND: PYTHON(./temp/function.py "1a" "b2" "5c" "a1b2c3" "123abc" "abcdef" "123456")