Fractional Function which f(f(x)) = x

Algebra Level 5

Let $a, b, c, d$ be non-zero real numbers such that the function $f(x) = \frac{ ax+b} { cx+d}$ defined on $\mathbb{R} \backslash \{ - \frac{d}{c} \}$ has the following properties:
1) $f(19) = 19$
2) $f(97) = 97$
3) $f(f(x) ) = x$

Suppose that there is a unique number $\alpha$ such that $\alpha \neq f(x)$ for any real number $x$ . What is the value of $\alpha$ ?