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!


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