The function takes a string input and rearranges the characters by moving the first character to the middle position, the middle character to the last position, and the last character to the first position.

[CODE]: 

```python
def approx_function(string: str) -> str:
    length = len(string)
    mid = length // 2
    return string[-1] + string[1:mid] + string[0] + string[mid + 1:-1] + string[mid]
```

This Python code snippet approximates the behavior of the given function based on the input-output pairs observed.