given f and the function composition fog, find g

Question

someone just gave me a question to solve and i've been wondering how to solve it: given f\begin{bmatrix}1&2&3&4\\2&3&4&1\end{bmatrix} and fog\begin{bmatrix}1&2&3&4\\3&1&4&2\end{bmatrix} find g=? i'd appreciate your help. I can find it by guessing but i'm looking for the exact solution. Thanks

Answer 1

I'm probably not going to go through every element, but let's consider the mapping of 1. We know that $f^{-1}(3) = 2$ and the $(f(g(1)) = 3$. So it seems that we'll need $g(1)$ to be 2, as we will then have $f(g(1)) = f(2) = 3$.

You can follow this line of reasoning to find the rest of g in the exact same way

Answer 2

let F: y -> z and G x -> y FoG(x)=z,let invF be the inverse function of F.

invFoFoG=(invFoF)oG=G.

G=invF(z)

invF is the following mapping which the same as G:

1 -> 4

2 -> 1

3 -> 2

4 -> 3