Chapter 12.8, Problem 14E

To determine

To find: the first four iterates (if necessary, round to nearest hundredth) of the function using the given initial value.

Answer to Problem 14E

  $0,-1,0\text{ and }-1$

Explanation of Solution

Given information:

  $f\left(x\right)=x^2-1,x_0=-1$

Calculation:

The first four iterates of function using the given initial value can be evaluated as shown below.

  $\begin{array}{c}{x}_1=f\left(x_0\right)\\ ={\left(-1\right)}^2-1\left[\text{Put }{x}_0=-1\right]\\ =0\\ x_2=f\left(x_1\right)\\ ={\left(0\right)}^2-1\left[\text{Put }{x}_1=0\right]\\ =-1\\ x_3=f\left(x_2\right)\\ ={\left(-1\right)}^2-1\left[\text{Put }{x}_2=-1\right]\\ =0\\ x_4=f\left(x_3\right)\\ ={\left(0\right)}^2-1\left[\text{Put }{x}_3=0\right]\\ =-1\end{array}$

The four iterates of $f\left(x\right)$ are $0,-1,0\text{ and }-1$ .