# Regression based on function recurrence

There is a hypothetical machine which takes an integer $$x$$ and returns an integer $$y$$ such that $$y=F(x)+\varepsilon$$ where $$\varepsilon$$ is an integer.

It is known that the function is of the form $$F(\alpha x_1) + \alpha F(x_2) = F(F(x_1 + x_2)) \forall x_1, x_2 \in Z$$

We are building a machine learning model to figure out how the machine behaves. We perform $$N$$ trials. $$x_1, x_2, .. x_N$$ are the inputs to the machine and $$y_1, y_2,. y_N.$$ are the corresponding outputs. We are trying to minimise the Mean Squared Error while fitting.

What would be the output of the model given an integer $$x$$ after training?

Hint: Plug $$x1 = 0, x2 = n$$ and $$x1 = 1, x2 = n-1$$. Can you infer the functional form of $$F$$ from this?

Yes this is a homework question. This is what I have using the hints.

$$F(n) = \frac{F(1)-F(0)}{\alpha} + F(n-1)$$

I am not looking for solution but just for ideas. I have no idea how to deal with this problem. Thank you!