Let $f,g : \mathbb R \to\mathbb R$. Suppose that $f\circ g$ is a strictly monotone growing function and $f$ is a strictly monotone decreasing function.
Can I conclude from those details that $g$ is a strictly monotone decreasing function?
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When exists apply the monotone $f^{-1}$ to the monotone $f\circ g$. Relevant. – A.Γ. Dec 10 '16 at 17:21
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Without adding "strictly" in front of "monotone", then no. $f(x) = 1$ is both monotone increasing and monotone decreasing, and no matter what $g$ is, the same will apply to $f\circ g$.
Arthur
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Wherever mentioned "montone" I meant to say "strictly monotone". Edited accordingly. What would be the answer in that case? – Dec 10 '16 at 18:06