The function takes a string input, increments the ASCII value of its first character, and removes consecutive repeating characters from the remaining substring.

[CODE]:
```python
def approximate_fx(input_string):
    first_char = chr(ord(input_string[0]) + 3) # Assuming an increment of 3
    output_string = first_char
    for i in range(1, len(input_string)):
        if input_string[i] != input_string[i-1]:
            output_string += input_string[i]
    return output_string
```
Please note that this code approximates the behavior of f(x) but might not exactly replicate the original function, as the increment value of the first character is not definitively determined.