Iterated Functions

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A function and its first two iterates
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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