Geometric represntation of terms for reccurence sequence with $f(x)=\frac{-1}{2} x+3$?

Question

let f be a function defined over $\mathbb{R}$ as $f(x)=\frac{-1}{2} x+3$ and $(u_{n})$ a sequence defined by recurrence relation such that :$u_{n+1}=f(u_n)$ , my problem is the Geometric representation of terms of $u_n$ with $u_0=0$ , $ , u_1,u_2,u_3$ using $y=x$ , then i have come across a problem when i tried to do projection over Graph of $f$ for the term $U_1$ , I want how do i got this representation for these terms ?

Answer 1

The function $$f(x)=\frac{-1}{2} x+3$$ has a fixed point at x=2 and this fixed point is an attractor. That means if you start at a point $x_1$ near the fixed point x=2 and iterate the function, then the sequence, $$ x_1, f(x_1), f(f(x_1)), ....$$ will approach $x=2$.

The graph that shows this process is the "Cubwebb" graph.

You start at $x_1$ and go vertical to hit the graph of $y=f(x)$ at the point $(x_1, f(x_1))$.

Then you go horizontal to hit the graph of $y=x$. Then you go vertical to hit the graph of $ y=f(x)$.

As you continue this process, your points are approaching the attractor.