# Iterative Nightmare

Calculus Level 1

Let $$f,g : \mathbb{R} \longrightarrow \mathbb{R}$$ such that $$f(2x)=2f(x)$$ and $$g(x) = x+ f\big(g(x)\big).$$

If $$2g(x)$$ is in the range of $$g$$ for all real $$x,$$ then is it always true that $$g(2x)=2g(x)?$$

Note: $$f$$ and $$g$$ are not necessarily continuous over $$\mathbb{R}.$$

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