The function f(x) takes a string input and returns the input unchanged if it contains uppercase letters, numbers, or special characters. If the input is a lowercase string, it either returns the same string or a transformed string according to a specific pattern.

[CODE]:
```python
def approx_function(input_string):
    if any(c.isupper() or not c.isalpha() for c in input_string):
        return input_string
    else:
        transformed_string = transform(input_string)  # Replace transform() with the specific transformation pattern
        return transformed_string
```
Note: You would need to replace `transform()` with the specific transformation pattern observed, as I cannot determine the exact transformation from the provided tests.