The function duplicates the first character of the input string and replaces 'o' with 'X', 'a' with 'P', 'i' with '`', 'g' with 'Y', and 'n' with 'Z'.

[CODE]:
```python
def approx_function(input_string):
    input_string = input_string[0] + input_string
    input_string = input_string.replace("o", "X")
    input_string = input_string.replace("a", "P")
    input_string = input_string.replace("i", "`")
    input_string = input_string.replace("g", "Y")
    input_string = input_string.replace("n", "Z")
    return input_string
```

Please note that this approximation might not cover all the edge cases and may need further refinement with additional testing.