The function replaces each alphabetical character in the input string with a corresponding character in the alphabet, while keeping the digits and some special characters unchanged. It applies the same transformation to both uppercase and lowercase letters. Specifically, it maps the 'A' to 'f', 'B' to 't', 'C' to 's', and so on, as shown in the response.

[CODE]:
```python
def approximated_function(input_string):
    original_alphabet = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
    transformed_alphabet = "ftshrbklopegcnzujqixaywmdvFTSHRBKLOPEGCNZUJQIXAYWMDV"

    output_string = ""
    for char in input_string:
        if char in original_alphabet:
            index = original_alphabet.index(char)
            output_char = transformed_alphabet[index]
        else:
            output_char = char

        output_string += output_char

    return output_string
```