Chapter 14.3, Problem 31E

To determine

To calculate: The first three iterates of the function $f(x)=0.5x-1$ using $x_0=8.$ .

Answer to Problem 31E

The first three iterations are $3,0.5,\text{ and }(-0.75).$

Explanation of Solution

Given information:

  $f(x)=0.5x-1$

Formula used:

Verhulst population model :- The model uses the recursive formula $p_{n+1}=p_n+r{p}_n\left(1-p_n\right),$ where n represents the number of time periods that have passed , $p_n$ represents the percent of the maximum sustainable population that exists at time n, and r is a growth factor.

Steps:- To iterate the function $f(x)$

1. Find the value of $f\left(x_0\right)$ of the initial value $x_0$ .

2.The second iterate $f\left(f\left(x_0\right)\right).$

And so on.

Calculation:

Consider,

$f(x)=0.5x-1$

and $x_0=8.$

Therefore, the three iterates are:-

  $\begin{array}{l}f(x_0)=0.5x_0-1\\ f(8)=0.5\left(8\right)-1\\ =4-1\\ =3\end{array}$

  $\begin{array}{l}f(f(x_0))=f(3)\\ f(3)=0.5\left(3\right)-1\\ =1.5-1\\ =0.5\end{array}$

  $\begin{array}{l}f(0.5)=0.5\left(0.5\right)-1\\ =0.25-1\\ =-0.75\end{array}$

Hence , the first three iterations are $3,0.5,\text{ and }(-0.75).$