All the details are attached below as .png images, thank you.

Directions: In this portfolio, you will use repeated function composition to explore

elementary ideas that are used in the mathematical field of chaos theory. Items

under the Questions headings will be submitted to your teacher as part of your

portfolio assessment. For all questions, make sure to be complete in your

responses. This can include details such as the function being iterated, the initial

values used, and the number of iterations. The phrase many iterations is used in

some of the questions. Interpret that to mean using enough iterations so that you

can come to a conclusion. If necessary, round decimals to the nearest ten-

thousandth.

Introduction

In this unit, you learned how to use function operations. One of the most important

operations is function composition. Just as two functions, f and g, can be composed

with each other, a function, f, can be composed with itself. Everytime that a

function is composed with itself, it is called an iteration. Iterations can be noted

using a superscript. You can rewrite (f•f)(x) as f'(x), (f•f•f)(x) as f'(x), and

so on. For this work, it is recommended that you use technology such as a graphing

calculator.

Example 1

Start with the basic function f(x) = 2x. If you have an initial value of 1, then you

end up with the following iterations.

f (1)=2-1= 2

f’(1) = 2.2.1=4

f'(1) = 2.2.2-1=8

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Questions

1. If you continue this pattern, what do you expect would happen to the

numbers as the number of iterations grows? Check your result by conducting

at least 10 iterations.

2. Repeat the process with an initial value of -1. What happens as the number

of iterations grows?

1

For this example, use the function f(x)==x+l and an initial value of 4. Note that

=+*+

with each successive iteration, you can use the previous output as your new input

to the function.

f(4)=:4+1=3

8°(4) = f(3) = 3.3 +1= 2.5

f°(4) = f(2.5) = :2.5+1 = 2.25

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Questions

3. What happens to the value of the function as the number of iterations

increases?

4. Choose an initial value that is less than zero. What happens to the value of

the function as the number of iterations increases?

5. Come up with a new linear function that has a slope that falls in the range

-1