Algebra Chaos Theory Worksheet Exercise Help

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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
.
.
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
.
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