The function seems to change every second character and append a new character at the end of the input string. The changes in the characters are:
- If the second character is 'a', 'b', or 'c', it replaces them with 'u', 'u', or 't', respectively.
- If the second character is 'e', it replaces it with 'n'.
- If the second character is 'o', it replaces it with 's'.
- If the second character is '1', '2', or '3', it replaces them with '2', '3', or 'w', respectively.

The added character at the end of the input string differs depending on the original string's length:
- If the string length is odd, the new character is 't'.
- If the string length is even, the new character is 'u', 't', or 'w' based on the last character in the input string.

[CODE]:
```python
def approximate_function(input_string):
    modified_string = ""
    for i, char in enumerate(input_string):
        if i % 2 == 1:
            if char == 'a':
                modified_string += 'u'
            elif char == 'b':
                modified_string += 'u'
            elif char == 'c':
                modified_string += 't'
            elif char == 'e':
                modified_string += 'n'
            elif char == 'o':
                modified_string += 's'
            elif char == '1':
                modified_string += '2'
            elif char == '2':
                modified_string += '3'
            elif char == '3':
                modified_string += 'w'
            else:
                modified_string += char
        else:
            modified_string += char

    if len(input_string) % 2 == 0 and input_string[-1] in ['a', 'b', 'c']:
        modified_string += 'u'
    elif len(input_string) % 2 == 0 and input_string[-1] == '1':
        modified_string += 'w'
    else:
        modified_string += 't'
    
    return modified_string
```
Keep in mind that this code approximates the observed behavior of the function but may not cover all possible input cases.