The function translates each input string's alphabetical characters by performing a simple cipher shift for the letters, leaving numerical and special characters unchanged. For each letter in the string, it performs the same shift operation regardless of the case (uppercase or lowercase). The shift seems to be consistent among the same characters from different input strings.

[CODE]:
```python
def approximate_function(input_string):
    shift = {
        'a': 'i', 'b': 'j', 'c': 'q', 'd': 'r', 'e': 't', 'f': 'u', 'g': 'v', 'h': 'w',
        'i': 'x', 'j': 'y', 'k': 'z', 'l': 'a', 'm': 'b', 'n': 'c', 'o': 'd', 'p': 'e',
        'q': 'h', 'r': 'k', 's': 'l', 't': 'm', 'u': 'n', 'v': 'o', 'w': 'p', 'x': 's',
        'y': 'f', 'z': 'g', 'A': 'I', 'B': 'J', 'C': 'Q', 'D': 'R', 'E': 'T', 'F': 'U',
        'G': 'V', 'H': 'W', 'I': 'X', 'J': 'Y', 'K': 'Z', 'L': 'A', 'M': 'B', 'N': 'C',
        'O': 'D', 'P': 'E', 'Q': 'H', 'R': 'K', 'S': 'L', 'T': 'M', 'U': 'N', 'V': 'O',
        'W': 'P', 'X': 'S', 'Y': 'F', 'Z': 'G'
    }

    output = ""
    for char in input_string:
        if char.isalpha():
            output += shift[char]
        else:
            output += char
    return output
```