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^2(1) = 2 * 2 * 1 = 4
· f^3(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. As the number of iterations grows, the numbers will continue to increase exponentially. After conducting 10 iterations with an initial value of 1, the results would be:
· f^4(1) = 2 * 2 * 2 * 2 * 1 = 16
· f^5(1) = 2 * 2 * 2 * 2 * 2 * 1 = 32
· f^6(1) = 2 * 2 * 2 * 2 * 2 * 2 * 1 = 64
· f^7(1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 128
· f^8(1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 256
· f^9(1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 512
· f^10(1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 1024
The numbers grow rapidly with each iteration, doubling each time.
2. When repeating the process with an initial value of -1, the numbers will also double with each iteration, but alternate between positive and negative values. After 10 iterations, the results would be:
· f(-1) = 2 * (-1) = -2
· f^2(-1) = 2 * 2 * (-1) = 4
· f^3(-1) = 2 * 2 * 2 * (-1) = -8
· f^4(-1) = 2 * 2 * 2 * 2 * (-1) = 16
· f^5(-1) = 2 * 2 * 2 * 2 * 2 * (-1) = -32
· f^6(-1) = 2 * 2 * 2 * 2 * 2 * 2 * (-1) = 64
· f^7(-1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * (-1) = -128
· f^8(-1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * (-1) = 256
· f^9(-1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * (-1) = -512
· f^10(-1) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * (-1) = 1024
The numbers grow rapidly and alternate between positive and negative values, doubling each time.