The function appends the previous character in the alphabet, or the previous digit, to the input. The alphabet wraps around from 'a' to 'z' and '1' to '0'. This applies to both lowercase and uppercase characters.

[CODE]:
```python
def approximate_function(input_str):
    result = ""
    for char in input_str:
        if char.isalpha():
            if char.lower() == 'a':
                result += char + 'z'
            else:
                result += char + chr(ord(char) - 1)
        elif char.isdigit():
            if char == '0':
                result += char + '9'
            else:
                result += char + str(int(char) - 1)
        else:
            result += char
    return result
```