The function takes input strings and outputs a new string by shifting its characters based on the distance of the input character from the first character in its corresponding category (uppercase letter, lowercase letter or digit).

[CODE]:
```python
def approx_f(x):
    result = ""
    for c in x:
        if c.isupper():
            shift = ord(c) - ord('A')
        elif c.islower():
            shift = ord(c) - ord('a')
        elif c.isdigit():
            shift = 0
        else:
            shift = 0
        
        result += chr(ord(c) + shift)
    return result
```
Please note that this is an approximation and may not cover all special characters.