The function appears to process each character in the input string with the following rules:
1. If the character is a lowercase 'e', change it to 'N'
2. If the character is an uppercase 'B', change it to 'N'
3. If the character is a lowercase 'o', change it to 'N'
4. If the character is an uppercase 'C', change it to 'N'
5. If the character is a lowercase 'a', change it to 'N'

The function then appends 'yNNh' at the end of the transformed string.

[CODE]:
```python
def approximate_function(input_string):
    output_string = ""
    for char in input_string:
        if char == 'e':
            output_string += 'N'
        elif char == 'B':
            output_string += 'N'
        elif char == 'o':
            output_string += 'N'
        elif char == 'C':
            output_string += 'N'
        elif char == 'a':
            output_string += 'N'
        else:
            output_string += char
    return output_string + 'yNNh'
```

Please note that this approximation might not cover all cases, but the provided inputs and outputs lead to these observations.