The function takes a string input and returns the middle character(s) of the input string. If the string length is odd, it returns one middle character, and if the length is even, it returns two middle characters.

[CODE]: 
```python
def approximate_function(input_string):
    length = len(input_string)
    middle = length // 2
    if length % 2 == 0:
        return input_string[middle - 1: middle + 1]
    else:
        return input_string[middle]
```

To make sure our approximation is correct, let's test the function again with various inputs.

COMMAND: PYTHON(./temp/function.py "testing" "functionality" "input" "output" "example")