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Let $h$ be the periodic function with period $2$ such that $h(x)=2|x|$ if $|x|\le1$. Let $f(x):=x+h(x)$ for all real $x$. Then the function $f$ is of degree $2$. However, if we had $f=P\circ g$ for some polynomial $P$ of degree $2$ and some increasing function $g$, then $f$ would switch at most once on $\mathbb R$ from increasing to decreasing, or vice versa -- whereas $f$ actually has infinitely many such switches.