Chapter 3.2, Problem 17E

Let A denote the set $R\backslash \left\{0,1\right\}$. Let i denote the identity function on A and define the functions $f,g,h,s,r: A\to A$ by

$f\left(x\right)=1-\frac{1}{x},g\left(x\right)=\frac{1}{1-x},h\left(x\right)=\frac{1}{x},r\left(x\right)=\frac{x}{x-1},s\left(x\right)=1-x$.

1. Show that $f\circ g=i\text{ and g}\circ r=s$. Complete the table, thereby showing that the composition of any two of the given functions is one of the given five, or the identity.

   

2. Which of the given six functions have inverses? Find (and identify) any inverses that exist.