Problem 110 - College Algebra Essentials

Question

In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a function to model data from 1980 through 2005 . The independent variable in my model represented the number of years after \(1980,\) so the function's domain was \(\\{x | x=0,1,2,3, \dots, 25\\}\).

Step-by-Step Solution

Answer

Yes, the statement makes sense as the domain of the function correctly represents the years from 1980 to 2005 in terms of number of years after 1980.

Step 1: Understanding the statement and context

The statement mentions a function which models data from the year 1980 through 2005. The independent variable in the model represents the number of years after 1980. We need to understand the question in terms of mathematical modeling and domain of a function.

Step 2: Evaluating the logic of the statement

Given that the independent variable represents the number of years after 1980, year 1980 will be represented as 0, year 1981 as 1 and so on. Hence the domain of the function, which is the set of possible values that the independent variable can have, shall be from 0 (representing 1980) to 25 (representing 2005). Therefore, the domain for this function is correctly represented by the set \(\{x | x=0,1,2,3, \dots, 25\}\).

Step 3: Providing a Conclusive Explanation

The given statement does make sense because the domain of the function correctly represents all the years from 1980 through 2005 in terms of number of years after 1980. The logic and setup of the function model align correctly with the mathematical principles of function modeling.

Key Concepts

Understanding the Function Domain

To grasp the concept of a function domain, it's helpful to think of it as the collection of all possible input values for which the function is defined. In mathematical terms, the domain specifies the range of values that the independent variable, often noted as \( x \), can take.
In the context of this exercise, the function models data from the years 1980 through 2005. Here, the domain is represented by the set \( \{ x | x = 0, 1, 2, 3, \ldots, 25 \} \).
This means the independent variable can take any value from 0 to 25, where each number corresponds to a year, from the initial year 1980 (\( x = 0 \)) to 2005 (\( x = 25 \)).
This set of values defines the boundaries within which the function operates. It ensures the function logically represents the timeframe in question.

Identifying the Independent Variable

The independent variable is a fundamental concept in mathematical modeling and serves as the foundation for analyzing data relationships. Essentially, it reflects the input value that can be changed freely in a function, directly affecting the dependent variable or output.
In this scenario, the independent variable is the number of years after 1980. Each increment by 1 in the independent variable corresponds to the passage of one year.

• For example, \( x = 0 \) represents the year 1980
• \( x = 1 \) corresponds to 1981
• Increasing further, \( x = 25 \) indicates the year 2005

This setup lets us understand how the passage of time impacts the model's output, which could be anything from population changes, economic data, or other time-dependent metrics.

The Role of Year Representation

Year representation is an essential part of creating a clear, effective function model, especially when dealing with time-series data. By turning years into a sequence of integers beginning at zero, we simplify the input for computational purposes, making it more manageable.
This representation allows us to uniquely map each year to a specific value of \( x \) in the domain, starting from zero for 1980. By using integers, we're simplifying data entry and calculations.

• It avoids confusion over specific yearly data points
• It provides uniform, simplified inputs for modeling functions

The advantage is a streamlined analysis of trends over time, which is crucial for understanding long-term patterns and dynamics.

Evaluating a Function Model

Function model evaluation involves checking the accuracy and logic of the model used to describe the real-world data. It ensures that inputs, outputs, and relationships within the model correctly reflect the scenario being analyzed.
In our example, the evaluation process confirms whether the domain correctly covers all years in the timeframe from 1980 to 2005. Fortunately, the model's logic aligns perfectly as the domain is from \( x = 0 \) to \( x = 25 \), matching each year to its corresponding integer.

• Each year is accurately mapped
• The function accurately covers the specified date range

This careful setup shows attention to detail and confirms the model's robustness, making sure it correctly represents the timeframe and can be reliably used for analysis.