Problem 110 - Precalculus Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning.I used a function to model data from 1990 through 2015 . The independent variable in my model represented the number of years after \(1990,\) 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 number of years after 1990 through to 2015.

Step 1: Cornerstone Concept

Recognize that the independent variable here represents the number of years after 1990. This means for the year 1990, x=0; for 1991, x=1; and so on.

Step 2: Matching the Range of Years with the Domain

Count the number of years from 1990 through 2015. This translates to a total of 26 years (including both 1990 and 2015). This range of years can be represented as follow: 0 (for 1990), 1 (for 1991), 2 (for 1992), ..., 25 (for 2015).

Step 3: Evaluating the Statement

Compare the domain provided in the statement with the actual calculated domain above. Here, they match perfectly. Hence, it makes sense as the years from 1990 through 2015 perfectly map to the domain \(\{x | x=0,1,2,3, \dots, 25\}\).

Key Concepts

Domain and Range

When dealing with functions, it's crucial to understand the concepts of domain and range. The domain of a function is the set of all possible input values that the function can accept. It is essentially "the independent variable values" that you can feed into the function. In our given exercise, the domain is { \( \{x | x=0,1,2,3, \dots, 25\} \)}, which means that the independent variable, representing years, ranges from 0 to 25. That indicates it's from 1990 (where \(x = 0\)) to 2015 (where \(x = 25\)).

• Understanding the domain helps you to know the timeline you're dealing with.
• It sets the boundaries for analyzing or predicting patterns over a period of time.

On the other hand, the range of a function is the set of all possible output values it can produce. Although not explicitly mentioned in this exercise, the range would relate to the data outcomes from the function model. Knowing both the domain and range helps map how the inputs affect the outputs.

Independent Variable

In mathematical modeling, the independent variable is a key concept. It is the value that you change or control directly, influencing the dependent variable. Here, the independent variable is the number of years after 1990, marking how many years have elapsed from that point. The statement in the exercise convinces us that modeling time periods often involves calculating the passage of years in precise steps.

• Example: If \(x = 5\), it represents the year 1995 (5 years after 1990).
• Helps in setting up equations to model real-world data efficiently.

Accurate identification and use of the independent variable are essential for valid and reliable predictions in mathematical models.

Mathematical Modeling

Mathematical modeling uses math to represent, analyze, make predictions, or simulate real-world phenomena. It is like crafting a blueprint of reality in mathematical terms. By employing mathematical models, you can extrapolate the data from various points, like in this exercise, covering the years from 1990 through 2015.
Here’s what makes effective mathematical modeling useful:

• Provides clearer insight into complex systems or patterns.
• Helps to test hypotheses and validate data-driven decisions.

In our scenario, the model leverages the independent variable to link years with data outcomes. For students, comprehending this concept can transform how they approach and solve problems across domains, from economics to biology.