Self-Inverse Function?

The condition $\large f(f(x)) = x$ tells us that the function is self-inverse , that is, $\large f^{-1}(x) = f(x)$ . Inverting $\large f(x) = \frac{ax-b}{cx}$ gives us $\large f^{-1}(x) = \frac{-b}{cx}$ . Matching coefficients in these rational functions gives us the condition $\large a = 0$ .

The two fixed points of $\large f(x) = \frac{-b}{cx} \$ , $\large x = 1$ and $\large x = 5$ , leads to $\large -2.5 = \frac{-b}{c}$ and $\large -0.5 = \frac{-b}{5c}$ , produces the system of equations $\large b = 2.5 c$ .

Thus, our self-inverse function is,

$\large f(x) = \frac{-2.5c}{cx} \ = \ -\frac{5}{2x}$

Now we can easily find $f(-20)$

$\large f(-20) = -\frac{5}{2(-20)} = \frac{-5}{-40} = \color{#D61F06}{\boxed{0.125}}$