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
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.
f (1) = 2 ⋅1 = 2
f 2 (1) = 2 ⋅2 ⋅1 = 4
f 3 (1) = 2 ⋅ 2 ⋅ 2 ⋅ 1 = 8
f4(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 16
f5(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 32
f6(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 64
f7(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 128
f8(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 256
f9(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 512
f10(1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 1 = 1024
2.Repeat the process with an initial value of −1. What happens as the number of iterations grows?
f (-1) = 2 ⋅(-1) = -2
f 2 (-1) = 2 ⋅2 ⋅(-1) = -4
f 3 (-1) = 2 ⋅ 2 ⋅ 2 ⋅ (-1) = -8
f4(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ (-1)= -16
f5(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ (-1) = -32
f6(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅(-1) = -64
f7(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ (-1) = -128
f8(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅(-1) = -256
f9(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅(-1) = -512
f10(-1) = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ (-1) = -1024
As you can see I've already done the conducting 10 iterations part for numbers 1 and 2, however my question would be : How do i explain to answer the question in number 2 "What happens as the number of iterations grows?" .
the values are correct.
As the number of iterations grows, so does the f value, either positive or negative.
And that's 2^x
2x means 2 times x: 2,4,6,8,...
also, did i conduct the 10 iterations correctly?
Thanks Steve! :)
This was my answer when I did the portfolio and I got it right😇
f(-1)=2(-1)=-2
f^2(-1)=f(-1)*f(-1)=-2*-2=4
f^3(-1)=f^2(-1)*f(-1)=4*(-2)=-8
f^4(-1)=f^3(-1)*f(-1)=-8*(-2)=16
f^5(-1)=f^4(-1)*f(-1)=16*(-2)=-32
As you can see the magnitude of the number increases, by being multiplied by 2 every time. Also as you can see the signs are altered, from negative to positive.
Well, as the number of iterations grows, it appears that the numbers keep doubling, but with a negative sign when the initial value is -1. It's like a twisted game of hide and seek - the numbers keep hiding behind negativity! So, as you keep going, the numbers become more and more negative, but their absolute values keep doubling. It's like being stuck in a never-ending cycle of negativity multiplied by 2. It's quite the wild ride!
To explain what happens as the number of iterations grows for question 2, you can observe the pattern in the values obtained.
In question 2, we started with an initial value of -1 and applied the function f(x) = 2x for each iteration.
As the number of iterations grows, the absolute value of the numbers obtained doubles with each iteration. However, the sign of the number alternates between positive and negative.
For example, f(-1) = -2, f2(-1) = 4, f3(-1) = -8, and so on.
This pattern suggests that as the number of iterations grows, the absolute value of the numbers becomes larger and larger, while the sign alternates between positive and negative.
In other words, the numbers oscillate between positive and negative values, with the magnitude doubling with each iteration.