Question

1. find the first 3 iterates of the function (f(x) = 0.75x) when (x_0 = 5).

(a) (6.67, 8.89, 11.85)
(b) (3.75, 2.8125, 2.109375)
(c) (5, 3.75, 2.8125)
(d) (4.25, 3.5, 2.75)

Explanation:

🆕 New Concept Discovered: Function Iteration
Applying a function repeatedly to its own outputs.

Step 1: Find the first iterate \( x_1 \)

To find the first iterate \( x_1 \), we plug the initial value \( x_0 = 5 \) into the function \( f(x) = 0.75x \):

$$ x_1 = f(x_0) = 0.75 \cdot 5 $$

$$ x_1 = 3.75 $$

Step 2: Find the second iterate \( x_2 \)

To find the second iterate \( x_2 \), we plug the output of the first step, \( x_1 = 3.75 \), back into the function:

$$ x_2 = f(x_1) = 0.75 \cdot 3.75 $$

$$ x_2 = 2.8125 $$

Step 3: Find the third iterate \( x_3 \)

To find the third iterate \( x_3 \), we plug the output of the second step, \( x_2 = 2.8125 \), back into the function:

$$ x_3 = f(x_2) = 0.75 \cdot 2.8125 $$

$$ x_3 = 2.109375 $$

Answer:

\( 3.75, 2.8125, 2.109375 \) (the second option)