Question 123
The actual functions of the real variable $f(x)$ of the form are considered: $f(x)=a x+b$, where $a$ being real numbers $b$. For what values of $a$ and $b$ is verified $f^{2000}(x)=x$ for all real numbers $x$? [Note: $f^{2}(x)=f(f(x))$, $f^{3}(x)=f(f(f(x)))$, and in general, $f^{n}(x)=f(f^{n-1}(x))=f(f(\ldots f(x)) \ldots)$ is defined ($n$ times)]