Another Functional Equation!

Algebra Level 5

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

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

Note - α,βN\alpha, \beta \in \mathbb N.

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


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