Another Functional Equation!

Algebra Level 5

\[\large{f(x+y) = f(x) + f(y) + \alpha}\]

Let \(f : \mathbb Q \to \mathbb Q\), where \(\mathbb Q\) is the set of all rational numbers, be such that the above functional equation holds true for all \(x,y \in \mathbb Q\). If \(f(\beta) = \alpha\), find the value of \( f(\alpha)\) when \(\beta=403, \alpha=2015\).

Note - \(\alpha, \beta \in \mathbb N\).

Bonus - Also generalize the value of \(f(\alpha)\) in terms of \(\alpha\) and \(\beta\).


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