Question 119
Polynomial $p(x) \in \mathbb{R}[x]$ is of odd degree $m$ greater than one. Also $f : \mathbb{R} \to \mathbb{Z}$ is a function such that for each real number $x$, we have $p(f(x)) = f(p(x))$.
a) Prove that the range of function $f$ is a finite set.
b) If $f$ is a non-constant function, prove that the equation $p(x) = x$ has at least two distinct real solutions.
c) Prove that for each natural number $n \gt 1$, there exist a function $f$ with a range containing exactly $n$ members and a polynomial $p(x)$ that satisfy all the aforementioned conditions.
a) Prove that the range of function $f$ is a finite set.
b) If $f$ is a non-constant function, prove that the equation $p(x) = x$ has at least two distinct real solutions.
c) Prove that for each natural number $n \gt 1$, there exist a function $f$ with a range containing exactly $n$ members and a polynomial $p(x)$ that satisfy all the aforementioned conditions.