Problem 53 - 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 \(2003 .\) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1991} & {1993} & {1995} & {1997} & {1999} & {2001} & {2003} \\ \hline 11 & {12} & {9} & {6} & {3} & {7} & {9} \\\ \hline\end{array} $$ Is the relation a function? Explain your reasoning.

Step-by-Step Solution

Answer

Yes, the relation is a function because each year maps to exactly one number of representatives.

Step 1: Define Function

A relation is a function if every input corresponds to exactly one output. In this context, evaluate if each year (input) is associated with a unique number of representatives with 30 or more years of service (output).

Step 2: Identify Inputs and Outputs

Inspect the table for the given data. The years are the inputs: \(1991, 1993, 1995, 1997, 1999, 2001, 2003\), and the corresponding number of representatives are the outputs: \(11, 12, 9, 6, 3, 7, 9\).

Step 3: Determine Uniqueness of Outputs

Verify if each year maps to a unique value. Check if any year might map to more than one value by reviewing the data column for outputs. Here, each year indeed maps to exactly one unique number of representatives.

Step 4: Conclude Relation is a Function

Since each year, which is the input, maps to exactly one output namely the number of representatives for that input, the relation satisfies the definition of a function.

Key Concepts

Input and Output Relation

In mathematics, the concept of input and output is fundamental to understanding how functions work. When we talk about an input, we are referring to any value that we can put into a relation or function. An output, on the other hand, is what we receive after inputting a value into the function.
For example, consider a relation where each year is an input, and the number of representatives with 30 or more years of service is the output. This means if you provide the year as input, the function (or relation) gives you the number of long-serving representatives.

• Input: Year (like 1991, 1993, etc.)
• Output: Number of Representatives (like 11, 12, etc.)

The input-output relationship is essential because it allows us to determine if something is a function. A function establishes a consistent method to determine outputs based on specific inputs.

Mapping

Mapping in mathematics refers to the way we associate inputs to outputs. In the context of functions, we often use the term 'mapping' to describe how one set of values relates to another.
Imagine a map as a rule book that specifies which inputs (years) correspond to which outputs (number of representatives). To identify if a relation is a function, we ensure the mapping is consistent for each pair of input and output.
In the given exercise, each year from the table uniquely maps to a specific number of representatives. This type of mapping means every input year corresponds with just one output number.
Mapping ensures clarity in functions:

• Each input must consistently lead to a specific output.
• No input has more than one output.

When all these conditions are met, we conclude that we do have a proper function.

Uniqueness in Functions

The principle of uniqueness is critical in defining a mathematical function. Uniqueness in this context means each input in a function can map to one and only one output. If an input could map to more than one possible result, the relation would not qualify as a function.
In the provided table problem, we are determining if each year uniquely maps to a specific number of representatives. Upon looking at the data:

• 1991 corresponds to 11 representatives.
• 1993 corresponds to 12 representatives.
• And so on.

Each year has a distinct output value, confirming that there is a unique correspondence between each year (input) and the respective count of representatives (output). Thus, the relation satisfies the function's condition of uniqueness.

Mathematical Tables

Mathematical tables provide a useful way to illustrate relationships between different sets of values. They are especially valuable in identifying patterns and determining functions as they allow easy examination of inputs and their corresponding outputs.
When dealing with tables, interpreting data becomes simpler; each column can represent a different set of values. In the exercise, the table clearly shows the relationship between the years and the number of representatives:

• The first row contains the years (inputs).
• The second row displays the number of representatives (outputs) for those years.

Tables make it straightforward to assess whether each input matches with a single, unique output, a necessary criterion for defining a function. By using tables, students can easily see the organized data and better understand the consistency in the input-output relationship.