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Find out alpha to solve the FE

Source: Rioplatense L3 2023 #5

December 6, 2023
functionalgebra

Problem Statement

Let R+\mathbb{R}^{+} be the set of positive real numbers. Determine all non-negative real number α\alpha such that there exist a function f:R+→R+f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+} such that f(xα+y)=(f(x+y))α+f(y)f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y) for any x,yx,y positive real numbers.
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