Iterated Functions

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A function and its first two iterates
Iterated2.jpg
Fields: Algebra, Dynamic Systems, and Analysis
Image Created By: Anna
Website: [anna's computer ]

A function and its first two iterates

This picture is the plot of g\left(x\right)=3.7x(1-x), which is in blue, g\left(g(x)\right), which is in pink and which we create by plugging in 3.7x(1-x) in for x. The yellow line is  g\left(g(g(x))\right), were we plug in g\left(g(x)\right) for x.


A More Mathematical Explanation

The iterates of a function is the set of functions \{f(x),f(f(x)),f(f(f(x))),f(f(f(f(x)))),....\}, which we d [...]

The iterates of a function is the set of functions \{f(x),f(f(x)),f(f(f(x))),f(f(f(f(x)))),....\}, which we denote by \{f(x),f^2(x),f^3(x),...f^n(x)\}.

Now, let's take f(x)=x^2. Then we see that

f^2\left(x\right)=f(f(x))=(x^2)^2=x^4,

f^3\left(x\right)=f(f(f(x)))=f(f^2(x))=(x^4)^2=x^8,

and f^4\left(x\right)=f(f(f(f(x))))=f(f^3(x))=(x^8)^2=x^{16}.

We can keep doing this, and write down a general form that f^n(x)=x^{2^n}




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