For this example, use the function f(x)=1/2x+1 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)=1/2 • 4+1=3

• f^2(4)= f(3)= 1/2 • 3 +1 =2.5

•f^3(4) =f(2.5)=1/2 • 2.5 +1=2.25

A) What happens to the value of the function as the number of iterations
increases?

B). Choose an initial value that is less than zero. What happens to the value of the function as the number of iterations increases?

C) Come up with a new linear function that has a slope that falls in the range -1<m<0. Choose two different initial values. For this new linear function, what happens to the function’s values after many iterations? Are the function’s values getting close to a particular number in each case?

D) Use the function g(x)=-x+2 with initial values of 4, 2, and 1. What happens after many iterations with all three initial values? How do the results of all three iterations relate to each other?

Answer:

Step-by-step explanation:

A) As the number of iterations increases, the output values of the function approach a specific value. In this case, as the number of iterations increases, the values of the function approach 2.

B) If we choose an initial value less than zero, such as -2, the function's values will oscillate between positive and negative values as the number of iterations increases. Specifically, the values will approach negative infinity as the number of iterations increases.

C) Let's consider the function g(x)=-1/2x+2. Two different initial values could be x=2 and x=4. For x=2, the values of the function approach 1 as the number of iterations increases. For x=4, the values of the function approach 1.5 as the number of iterations increases.

The function's values are getting close to a particular number in each case. For both initial values, the values of the function approach 1 as the number of iterations increases.

D) Using the function g(x)=-x+2 with initial values of 4, 2, and 1, we get:

• g^2(4) = g(2) = -2+2 = 0

• g^2(2) = g(0) = -0+2 = 2

• g^2(1) = g(1) = -1+2 = 1

After many iterations, all three initial values converge to the same value of 1. The results of all three iterations are related in that they all approach the same value after many iterations.