Problem 19 - Algebra 2

Question

Find the first three iterates of each function for the given initial value. $$ f(x)=4 x-3, x_{0}=2 $$

Step-by-Step Solution

Answer

The first three iterates are 5, 17, and 65.

Step 1: Initialize the Iteration

To find the first three iterates of the function, we start with the initial value given. Here, the initial value is \( x_0 = 2 \).

Step 2: Compute the First Iterate

Substitute \( x_0 = 2 \) into the function \( f(x) = 4x - 3 \) to find \( x_1 \):\[x_1 = f(x_0) = 4 \times 2 - 3 = 8 - 3 = 5\]

Step 3: Compute the Second Iterate

Take the result from the first iterate, \( x_1 = 5 \), and substitute it back into the function to find \( x_2 \):\[x_2 = f(x_1) = 4 \times 5 - 3 = 20 - 3 = 17\]

Step 4: Compute the Third Iterate

Use the result from the second iterate, \( x_2 = 17 \), and substitute it back into the function to find \( x_3 \):\[x_3 = f(x_2) = 4 \times 17 - 3 = 68 - 3 = 65\]

Key Concepts

Iterative Process

In mathematics, an iterative process is a method of computing approximations to the solutions of problems. It involves repeating a series of steps to achieve desired results. This technique is commonly used in solving equations and analyzing functions.

In our context, an iterative process starts with a given initial value, and then the process of applying a function repeatedly follows. The outcome from one iteration becomes the input for the following one. This method is helpful for improving guesses or refining solutions over multiple stages. Here are key details:

• Iteration: Each cycle of computation in the process.
• Convergence: The goal of iterations is to get closer to a particular value or solution.
• Function application: The given function is applied repeatedly, providing iterative results.

Think of it like baking cookies. You repeat the baking steps for each batch, ensuring that each one comes out just right. The repeated steps help improve and maintain quality.

Initial Value

The initial value in an iterative process serves as the starting point. This value is crucial because it determines where you kick-off the iterative cycles. The performance and success of your iterations can depend significantly on this initial choice.

Here's why the initial value is important:

• Starting Point: It defines where your series of calculations begin.
• Accuracy: A well-chosen initial value can lead to faster convergence.
• Dependence: The outcome of the iterative process is often sensitive to this starting value.

In the exercise at hand, the initial value was given as 2, serving as the starting block for calculating subsequent iterates. Occasionally, finding the optimal initial value demands experimentation or prior computation from similar problems.

Algebraic Functions

Algebraic functions are expressions constructed using algebraic operations, like addition, subtraction, multiplication, division, and exponentiation involving variables. Understanding these functions is vital in algebra and broader mathematical contexts.

In our function iteration exercise, we used the algebraic function: \[ f(x) = 4x - 3 \]This type of function showcases simple linear relationships. Consider these points about linear algebraic functions:

• Form: Typically expressed as `ax + b` where `a` and `b` are constants.
• Application: Used extensively to model real-world linear relationships.
• Graph: Represents a straight line, where `a` is the slope.

The simplicity and directness of algebraic functions make them perfect for iterative calculations, as demonstrated with the exercise, by simply substituting and calculating each new iterate. This recurrent substitution illustrates algebra's power in breaking down complex problems.