The function appears to be performing a Caesar cipher, shifting each alphabetic character by 19 positions forward in the alphabet, while leaving numerical characters and special symbols unchanged. The shift operation wraps around the alphabet (e.g., 'z' maps to 's'). The function handles both uppercase and lowercase letters.

[CODE]:
```python
def approx_function_f(input_str):
    shifted_str = ""
    for char in input_str:
        if 'A' <= char <= 'Z':
            shifted_str += chr(((ord(char) - ord('A') + 19) % 26) + ord('A'))
        elif 'a' <= char <= 'z':
            shifted_str += chr(((ord(char) - ord('a') + 19) % 26) + ord('a'))
        else:
            shifted_str += char
    return shifted_str
```
This python code should approximate the behavior of the given function f(x).