SE-SKOLORNAS 1993 Problem 6

For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$ . For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$ , $f(x_2) = x_3$ and $f(x_3) = x_1$ ?