VN-TST 1995 Problem 6

Consider the function $f(x) = \frac {2x^3 - 3}{3x^2 - 1}$ .

$1.$ Prove that there is a continuous function $g(x)$ on $\mathbb{R}$ satisfying $f(g(x)) = x$ and $g(x) > x$ for all real $x$ .

$2.$ Show that there exists a real number $a > 1$ such that the sequence $\{a_n\}$ , $n = 1, 2, \ldots$ , defined as follows $a_0 = a$ , $a_{n + 1} = f(a_n)$ , $\forall n\in\mathbb{N}$ is periodic with the smallest period $1995$ .