Problem 52 - Algebra 2

Question

Government For Exercises \(50-53\) , use the table below that shows the number of members of the U.S. House of Representatives with 30 or more consecutive years of service in Congress from 1991 to \(2003 .\) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1991} & {1993} & {1995} & {1997} & {1999} & {2001} & {2003} \\ \hline 11 & {12} & {9} & {6} & {3} & {7} & {9} \\\ \hline\end{array} $$ Identify the domain and range. Determine whether the relation is discrete or continuous.

Step-by-Step Solution

Answer

Domain: \( \{1991, 1993, 1995, 1997, 1999, 2001, 2003\} \); Range: \( \{3, 6, 7, 9, 11, 12\} \); Relation is discrete.

Step 1: Identify the Domain

The domain refers to the set of all possible input values. Here, "x" values or years between 1991 and 2003 are given as 1991, 1993, 1995, 1997, 1999, 2001, and 2003. Therefore, the domain is \( \{1991, 1993, 1995, 1997, 1999, 2001, 2003\} \).

Step 2: Identify the Range

The range refers to the set of all possible output values. These are the number of U.S. House of Representatives members with 30 or more consecutive years of service for each year. From the table, these values are 11, 12, 9, 6, 3, 7, and 9. Therefore, the range is \( \{3, 6, 7, 9, 11, 12\} \).

Step 3: Determine if the Relation is Discrete or Continuous

A relation is discrete if the input values are distinct, individual points (such as specific years), rather than forming a continuum. In this problem, each year is distinct and separate as opposed to spanning a continuous spectrum of years. Therefore, the relation is discrete.

Key Concepts

Domain and Range

In discrete math, the concept of domain and range is crucial when analyzing relations and functions. A domain consists of all the possible input values that a relation or function can accept. In this context, viewing the year as an input, the domain represents the specific years for which data is available. Each year can be thought of as an independent entity or 'input' value.

Let's break it down with our exercise example. The data presents the years: 1991, 1993, 1995, 1997, 1999, 2001, and 2003. This set of years is the domain.

• Domain: Every year between 1991 to 2003 as presented in the data.

The range, on the other hand, contains all the resulting 'output' values tied to the domain inputs. In this exercise, the number of House of Representatives members, corresponding to each year, forms the range. These values alert us to how many members reached 30 years or more of consecutive service.

In summary:

• Range: The numbers 3, 6, 7, 9, 11, and 12, which indicate member counts.

Fully understanding domain and range in this context helps recognize the relationship between time-specific data (years) and outcomes (member counts).

Discrete Relation

Understanding whether a relation is discrete or continuous is another key part of discrete math. Relations can be classified based on the nature of their inputs and outputs.

In a discrete relation, the input values are individual, separate points. Think of them like distinct dots on a graph rather than a connected line. For instance, in our exercise, each data point corresponds to a specific year. The years do not form a continuous range, but rather a series of individual points. Because we only care about specific, listed years, this is not a continuous process spanning every year from one to the next.

• Discrete: Each year (e.g., 1991 or 1993) is a separate, independent data point.

The distinct nature of this data means our analysis focuses on individual years instead of creating connections or interpolations between these years. Identifying whether a relation is discrete helps in selecting appropriate analytical methods and correctly interpreting the data.

Year-based Data Analysis

Year-based data analysis involves examining data that is time-specific, each time point holding its own unique set of values or historical context. In our exercise, this data tracks the number of House members with significant tenure from 1991 through 2003.

Analyzing such data helps us understand trends over time, such as whether fewer members maintained long tenures in more recent years, or if patterns changed dramatically at any point. For this analysis to be fruitful, recognizing the discrete nature of the time points is pivotal as it emphasizes treating each year, not as a segment of a flowing timeline, but as a distinct snapshot of that moment's reality.

• Year By Year: Each year provides a standalone insight into the membership count, helping in observing any yearly changes.

Discrete relationships like these are common in historical data analysis, which often deals with year-by-year records, census data, or financial reports. Understanding this provides insight into not only past trends in the data but potentially guiding future forecasts based on identified patterns.