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 \( \alpha \)?