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

Algebra Level 5

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

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

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