Function of a function of a ...

When iterating the recursive equation you apply the function on the function on the ...

\[x,f(x),f(f(x)),f(f(f(x))),\ldots \]

At a fixpoint all the functions of a function of a ... intersect.

You can apply the same function on itself over and over again in GeoGebra by using the spread sheet.

• Enter the function in terms of x into A1. The cell A1 becomes the function A1(x).
• Now enter the function in terms of A1(x) into the cell A2.
• Click on the blue rectangle in the lower right corner of A2 and drag down in order to do a relative copy.

Exercise

Enter a slider a into GeoGebra. Draw graphs of a function of a function of a function...using sin(ax). Do not copy more than 6 or 7 functions (otherwise it will lag too much).

Note that for some values of a there are points of intersection that are not fixpoints. It is hence possible to have points of attraction to a recursive equation that are not fixpoints. Such points can give rise to chaos.