The function adds a 't' between every pair of consecutive characters if they are the same or both are alphabetic and have the same case. After that, the function appends the first character of the input string to the end of the output string.

[CODE]:
```python
def approximate_function(s):
    output = ""
    for i in range(len(s) - 1):
        output += s[i]
        if (s[i].isalpha() and s[i + 1].isalpha() and s[i].isupper() == s[i + 1].isupper()) or (s[i] == s[i + 1]):
            output += 't'
    output += s[-1]
    output += s[0]
    return output
```