Problem 21 - Algebra 2

Question

Find the first three iterates of each function for the given initial value. $$ f(x)=5 x+1, x_{0}=-1 $$

Step-by-Step Solution

Answer

The first three iterates are -4, -19, and -94.

Step 1: Calculate the First Iterate

To find the first iterate, substitute the initial value of \(x_0 = -1\) into the function \(f(x) = 5x + 1\). This yields: \[ f(x_0) = 5(-1) + 1 = -5 + 1 = -4. \] Thus, the first iterate \(x_1\) is \(-4\).

Step 2: Calculate the Second Iterate

Now, use the first iterate, \(x_1 = -4\), as the input into the function. Substitute \(-4\) into \(f(x)\): \[ f(x_1) = 5(-4) + 1 = -20 + 1 = -19. \] Thus, the second iterate \(x_2\) is \(-19\).

Step 3: Calculate the Third Iterate

Use the second iterate, \(x_2 = -19\), as the input into the function. Substitute \(-19\) into \(f(x)\): \[ f(x_2) = 5(-19) + 1 = -95 + 1 = -94. \] Thus, the third iterate \(x_3\) is \(-94\).

Key Concepts

Understanding Initial Value

When dealing with functions, an 'initial value' is a starting point from which calculations or iterations begin. Think of it as the initial condition or the starting input. In mathematics, knowing the initial value is crucial because it determines how the function will be evaluated and iterated.For the given exercise, the initial value is denoted as \( x_0 = -1 \). This means you begin your function iteration process with \(-1\) as the initial input. Knowing this helps in building the sequence of iterates or steps needed to evaluate the function repeatedly. By using this initial value, you can generate subsequent values—each representing an output based on the previous one. This concept is especially important in diverse areas like differential equations, simulation modeling, and computer algorithms that rely on similar iterative processes.

The Role of Function Evaluation

'Function evaluation' is the process of substituting a given input into a function to find the output. Essentially, it helps to understand how a function transforms an input value to an output value.In this example, you have the linear function \( f(x) = 5x + 1 \). Function evaluation begins by substituting \( x_0 = -1 \), the initial value, into the function. You then solve the expression:- First iterate: \( f(x_0) = 5(-1) + 1 = -4 \).- Second iterate: substitute the first iterate \( x_1 = -4 \) back into the function, yielding \( f(x_1) = 5(-4) + 1 = -19 \).- Third iterate: substitute the second iterate \( x_2 = -19 \) to get \( f(x_2) = 5(-19) + 1 = -94 \).This step-by-step function evaluation shows how the function processes the input to provide a new output each time. Understanding this iterative substitution helps in predicting the nature of the function's output based on its structure.

Exploring Linear Functions

Linear functions are a specific group of algebraic functions that have the form \( f(x) = mx + b \). Here, \(m\) and \(b\) are constants, where \(m\) represents the slope and \(b\) is the y-intercept. In our example, the function \( f(x) = 5x + 1 \) is linear.Linear functions are characterized by producing straight-line graphs when plotted on a coordinate plane. The constant slope \( m = 5 \) tells you the rate at which the function's output will increase or decrease for each unit increment in \( x \). Thus with each iteration:- For the first iterate, starting with \( -1 \), the changes follow the function's formula: multiplying by 5 and adding 1.- The constant nature of linear functions means the relationship between subsequent iterates remains consistent relative to their inputs.In practical terms, understanding linear functions is vital since they provide a straightforward way to predict relationships, explain trends, and solve problems related to change and rate. Their simplicity means they often serve as the foundation for understanding more complex mathematical functions.