[math-fun] Something I figured out

For 2 x 2 matrix M define F(M) = f : R->R : x => (M11 x + M12) / (M21 x + M22). Then F(AB) = F(A) o F(B) So the composition of unreduced order 1 rational functions is isomorphic to the product of 2x2 matrices. I assume this is well known, but I thought it was pretty cool. This means that finding, say, all order 1 rational functions f with f(f(x)) = x would reduce to finding all 2x2 matrices M with M^2 = I.