The function f(x) appears to perform a Caesar cipher shift on its input string by incrementing each character by 1 position. Additionally, it repeats the last character of the input string and appends it to the end of the output.

[CODE]: Here's a Python code that approximates the behavior of the function:

```python
def f_approx(x):
    output = ""
    for ch in x:
        if ch.isalpha():
            if ch.isupper():
                output += chr(((ord(ch) - 65 + 1) % 26) + 65)
            else:
                output += chr(((ord(ch) - 97 + 1) % 26) + 97)
        else:
            output += ch
    output += x[-1]
    return output
```
Keep in mind that this is an approximation and might not cover all edge cases present in f(x).