composite function by many "layers"

Since $gg$ is the identity map, $g^{80}$ is the identity map, so we just need to calculate $f^{20}$ . Now $f(x)-4 = 2(f-x)$ , so $f^n(x)-4=2^n(x-4)$ (by induction), and so $f^n(x) \; = \; 4\big[2^{n-2}x - (2^n-1)\big]$ Thus $a=2^{n-2}$ , $b=2^n-1$ and $c=\boxed{4}$ for any $n \ge 2$ . Since $b$ is odd, $a$ and $b$ are coprime.

c=4