Problem 18 - Algebra 2

Question

Find the first three iterates of each function for the given initial value. $$ f(x)=9 x-2, x_{0}=2 $$

Step-by-Step Solution

Answer

The iterates are 16, 142, and 1276.

Step 1: Initial Setup

We need to find the first three iterates of the function \( f(x) = 9x - 2 \) starting from the initial value \( x_0 = 2 \). This means applying the function iteratively starting with \( x_0 \).

Step 2: Calculate First Iterate: \( x_1 \)

To find \( x_1 \), we substitute \( x_0 = 2 \) into the function: \( x_1 = f(x_0) = 9(2) - 2 = 18 - 2 = 16 \). Thus, the first iterate \( x_1 = 16 \).

Step 3: Calculate Second Iterate: \( x_2 \)

Next, substitute \( x_1 = 16 \) into the function to find \( x_2 \): \( x_2 = f(x_1) = 9(16) - 2 = 144 - 2 = 142 \). Thus, the second iterate \( x_2 = 142 \).

Step 4: Calculate Third Iterate: \( x_3 \)

Finally, substitute \( x_2 = 142 \) into the function to find \( x_3 \): \( x_3 = f(x_2) = 9(142) - 2 = 1278 - 2 = 1276 \). Thus, the third iterate \( x_3 = 1276 \).

Key Concepts

Function Iteration

Function iteration involves applying a mathematical function repeatedly, starting from an initial value. It is commonly used to explore the behavior of functions over multiple steps, thus simulating the progression over time. When we iteratively apply a function like in the given exercise, we start with an initial value and substitute it into the function equation. This produces the first iterate, which is then used as the input for the next round of computation.
This process can be repeated as many times as needed to generate a sequence of values that provide insight into the function's pattern. Here are the general steps you should follow when iterating a function:

• Identify the initial value and the function that needs to be iterated.
• Substitute the initial value into the function to calculate the first iterate.
• Use each new iterate as the input for the next application of the function.
• Continue this process to generate as many iterates as required.

Understanding function iteration is vital in mathematics as it helps to predict outcomes, understand dynamic systems, and solve complex algebraic problems.

Sequence Generation

Sequence generation is the creation of a list of numbers generated by iteratively applying a defined rule or function. Each number, called an iterate, depends on the previous value and the specific function or formula being used. In the case of our exercise, we see a clear example of sequence generation through the repeated application of the linear function \( f(x) = 9x - 2 \).

• Starting with \( x_0 = 2 \), the sequence begins.
• The first iterate, \( x_1 = 16 \), is derived by inserting \( x_0 \) into the function.
• This process is repeated, using each new iterate as the input for the next step.

By doing this, we form a sequence \( x_0, x_1, x_2, x_3, \) and so on. As more iterates are calculated, patterns and behaviors emerge, offering insight into the characteristics and tendencies of the function being used. This is particularly useful in understanding how initial conditions affect future outcomes and how a function behaves over a series of iterations. Sequence generation can aid in many mathematical practices such as predicting growth trends and solving sequences in algebraic contexts.

Algebra Problem Solving

In algebra problem solving, iterative functions are crucial for finding numerical solutions and understanding functional behavior. The original problem exemplifies the application of algebraic manipulation to achieve a series of solutions, demonstrating an underlying mathematical pattern. Solving algebra problems often involves working through iterative procedures, thus improving clarity and predictability.
It is essential to:

• Understand the function involved and how to apply it to initial and subsequent values.
• Perform careful algebraic manipulations to avoid errors in calculation.
• Note the continuity in patterns produced through further iterates, reinforcing the understanding of sequences.

The exercise used a linear function to create a sequence, emphasizing consistent steps that yield predictable results. Applying algebra to solve problems like this requires careful consideration of function properties and formulas, thereby enhancing problem-solving skills across different branches of mathematics.