Powers of a mapping
Task number: 2812
Let \(f(x)=(1-x)^{-1}\). Determine \(f\circ f\) a \(f\circ f\circ f\).
Resolution
\( \displaystyle (f\circ f)(x)=f(f(x))=\frac{1}{1-f(x)}=\frac{1}{1-\frac{1}{1-x}}=\frac{x-1}{x}=1-\frac1x \)
\( \displaystyle (f\circ f\circ f)(x)=f((f\circ f)(x))=\frac{1}{1-\left[1-\frac1x\right]}=x \)
Result
Iterating the function yields \((f\circ f)(x)=1-\frac1x\) and \((f\circ f\circ f)(x)=x\).
