Problem 39 - College Algebra with Modeling and Visualization

Question

Modeling Decide if the situation could be modeled by a one-to-one function. The population of the United States from 1980 to 2000

Step-by-Step Solution

Answer

Yes, it can be modeled by a one-to-one function.

Step 1: Understand the Definition of One-to-One Function

A function is one-to-one if each input corresponds to exactly one unique output, and no output is repeated for different inputs. In mathematical terms, if \( f(a) = f(b) \) implies \( a = b \), then the function is one-to-one.

Step 2: Analyze Population Growth as a Function

Consider the population of the United States from 1980 to 2000 as a function, where time (in years) is the input and the population size is the output. In this function, each year in the range is paired with a specific population number.

Step 3: Check for Uniqueness of Output

Analyze whether each year maps to a unique population size. The population size generally grows or decreases over time without repeating exactly for a different year. If each year corresponds to a unique population size, this suggests a one-to-one relation.

Step 4: Historical Context of Population Data

Consider the historical data during the 1980 to 2000 period. The population of the United States was increasing each year, driven by factors like birth rates, decreasing mortality rates, and immigration. This historical context supports the idea that population size did not remain the same for different years.

Step 5: Conclusion on One-to-One Modelling

Since each year has a unique population, and no two years have the same population figure, the situation can be modeled by a one-to-one function. Thus, for each year (input), there is a unique population size (output), confirming it as one-to-one.

Key Concepts

Population Modeling

Population modeling involves examining how populations change over a period of time. In the context of the United States from 1980 to 2000, population modeling can be viewed as analyzing the growth trends and factors influencing the number of inhabitants each year. Utilizing a model helps in understanding trends and making predictions for future growth. Though various factors contribute to these trends, the main ones can include:

• Birth rates
• Mortality rates
• Immigration levels

All of these factors collectively contribute to the overall population changes. By modeling these changes, we can see if each year produces a unique population size, which is key to determining whether the relationship functions as a one-to-one correlation. If each year corresponds to a specific population size without any repetition, then the model reflects a one-to-one function as it captures the uniqueness of the population at each point in time.

Function Analysis

When analyzing functions, it is important to consider the relationship between the input and output values. With population function analysis, the 'input' could be the year, while the 'output' is the population for that specific year. The aim of function analysis is to determine how these inputs and outputs relate and if specific types of functions apply. For the population data between 1980 and 2000, one checks whether every year corresponds to a different population size. In such function analysis, the behavior of the data determines whether the function is:

• Injective (one-to-one)
• Surjective (onto)
• Bijective (both one-to-one and onto)

For the exercise, determining a one-to-one nature means verifying that no two different inputs (years) produce the same output (population size). This ensures that each year has a unique output value, conforming to the properties of one-to-one functions.

Mathematical Function Properties

Understanding mathematical function properties is crucial for identifying the nature of relationships modeled by a function. A one-to-one function, or injective function, has an important characteristic: each input has exactly one unique output. This is not just a theoretical property; it has practical implications in analysis and modeling.For a function to be one-to-one:

• If two outputs are equal, then the inputs must also be equal, which can be mathematically expressed as: if \( f(a) = f(b) \), then \( a = b \).
• The graph of the function will pass the horizontal line test, meaning no horizontal line will intersect the graph more than once.

In population modeling, applying these properties means examining whether each year yields a distinct population value. If true, it indicates that the population model respects the property of being a one-to-one function, making it a valuable tool in understanding how populations evolve over time.