The function takes a string input and increments the first alphabetical character to the next character in the alphabet. If the first character is not alphabetical, it remains unchanged. If the first alphabetical character is 'z' or 'Z', it wraps around to 'a' or 'A', respectively.

[CODE]: 
```python
def approximate_function(input_str):
    modified_str = list(input_str)
    for i, char in enumerate(modified_str):
        if char.isalpha():
            if char.lower() == 'z':
                modified_str[i] = 'a' if char.islower() else 'A'
            else:
                modified_str[i] = chr(ord(char) + 1)
            break
    return ''.join(modified_str)
```

Please let me know if you want to test the function on more inputs or need further explanation.