Chapter 3.2, Problem 11E

Let $S=\left\{1,2,3,4\right\}$ and define functions $f,g:S\to S$ by $f=\left\{\left(1,3\right),\left(2,2\right),\left(3,4\right),\left(4,1\right)\right\}$ and $g=\left\{\left(1,4\right),\left(2,3\right),\left(3,1\right),\left(4,2\right)\right\}$. Find

1. $g^{-1}\circ f\circ g$

2. $f\circ g^{-1}\circ g$

3. $g\circ f\circ g^{-1}$

4. $g\circ g^{-1}\circ f$

5. $f^{-1}\circ g^{-1}\circ f\circ g$