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anonymous
 4 years ago
Consider the continuous function
f(x)R:R>R
that satisfies
f(0)=0 and f(f(x))=2f(x)∀x∈R
Find f(x).
Problem taken from AoPS.
anonymous
 4 years ago
Consider the continuous function f(x)R:R>R that satisfies f(0)=0 and f(f(x))=2f(x)∀x∈R Find f(x). Problem taken from AoPS.

This Question is Closed

Mr.Math
 4 years ago
Best ResponseYou've already chosen the best response.1An obvious function that satisfies the given conditions is \(f(x)=2x.\) But is this the only such function?! I doubt that.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0That's why there should be a formally written proof. :3

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Assume \(f(x) \in \mathbb{R}[x].\) Let \(f(x)=a_0+a_1x+\dots+a_kx^k.\) \(f(0)=0\) implies \(a_0=0.\) \[ \begin{align} f(f(x))&=a_1(a_1x+\dots+a_kx^k)+\dots+a_k(a_1x+\dots+a_kx^k)^k\\ &=a_1(a_1x+\dots+a_kx^k)+a_2(a_1x+\dots+a_kx^k)^2\\ &+\dots+a_k(a_1x+\dots+a_kx^k)^k\\ &=2(a_1x+\dots+a_kx^k)\\ &\Rightarrow a_2,a_3,\dots,a_k=0\\ &\Rightarrow a_1^2x=2a_1x\\ &\Rightarrow a_1=2\\ \therefore f(x)&=2x \end{align} \]
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