So I read in another thread about involutory functions, he claims for any real numbers $a$ and $b$, the function: $$f(x) = a + \frac{b}{x-a} = \frac{ax + (b-a^2)}{x-a}$$ satisfies $$f(f(x)) = a + \frac{b}{a + \frac{b}{x-a} - a} = a + (x-a) = x$$
And well, I thought I'd give it a try. So I want to find values of $a, b$ and $c$ at which the function below is involutory:
$$f(x) = \frac{x-a}{bx-c}$$
And I of course want to solve for $a, b$ and $c$. Now, I don't consider myself good enough to know in what direction I need to go to solve this.
My attempt was $$f(f(x)) = \frac{\frac{x-a}{bx-c}-a}{b\cdot\frac{x-a}{bx-c}-c} = x$$
$$\iff\frac{\frac{x-a-abx+ac}{bx-c}}{\frac{bx-ba-cbx+c^2}{bx-c}} = \color{red}{\frac{x-a-abx+ac}{bx-ba-cbx+c^2} = x}$$
Now, how would I continue on to find values for $a,b$ and $c$, which satisfy the equation I colored red? By glance I happened to notice that if $a=b=0$ and $c=-1$, the equation is satisfied; but I want to be able to do it algebraically.
Any help is appreciated, thanks.
Edit 1. I was thinking about using a matrix; however I haven't learned about them yet, but I'd be happy to study a solution using them to see if I can get a hang of it.
http://www.wolframalpha.com/input/?i=x+%3D+%28x-a-abx%2Bac%29%2F%28bx-ba-cb*x%2Bc^2%29