My question is about existence of a non-trivial solution of the functional equation $f(f(f(x)))=-x$ where $f$ is a continuous function defined on $\mathbb{R}$. Also, what about the general one $f^n(x)=-x$ where $f^n$ is understood in the sense of composition of functions and $n$ is odd. Eventually, is there some theory about continuous solutions of $f^n(x)=g(x)$ where $g$ is a fixed continuous function. I tried a research here but all what I found was about $f^2(x)=g(x)$, i.e, "square root" in the sense of composition. Thanks.
EDITED : I was looking for a non-trivial solution with $n$ odd, sorry for the inconvenience. I reformulated my question. Thanks for the last poster who showed that the unique continuous solution to $f^n(x)=-x$ where $n$ is odd is the trivial one.