Problem 25 - Prealgebra

Question

Express each relation as a table and as a graph. Then determine the domain and range. $$\\{(4,5),(5,2),(1,6)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{1, 4, 5\}\); Range: \(\{2, 5, 6\}\).

Step 1: Understanding the Relation

We are given a set of ordered pairs: \(\{(4,5),(5,2),(1,6)\}\). Each pair consists of a first element (x-coordinate) and a second element (y-coordinate), which can be represented in relation form to understand the connection between the elements.

Step 2: Constructing the Table

To represent this relation as a table, create two columns: one for 'x' and one for 'y'. Populate the table with the given ordered pairs as follows: | x | y | |---|---| | 4 | 5 | | 5 | 2 | | 1 | 6 |.

Step 3: Graphing the Relation

To graph the relation, plot each ordered pair on a set of axes. - Plot (4, 5) by moving 4 units along the x-axis and 5 units up on the y-axis. - Plot (5, 2) by moving 5 units along the x-axis and 2 units up. - Plot (1, 6) by moving 1 unit on the x-axis and 6 units up. Each point on this graph represents an ordered pair from the relation.

Step 4: Determining the Domain

The domain is the set of all x-coordinates from the given pairs. Extract these from the table: 4, 5, and 1. Thus, the domain is \(\{1, 4, 5\}\) as a set, noting that sets do not repeat elements.

Step 5: Determining the Range

The range is the set of all y-coordinates from the given pairs. Extract these from the table: 5, 2, and 6. Thus, the range is \(\{2, 5, 6\}\) noting that sets list each element once without repetition.

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a group of two numbers written in a specific order within parentheses, like this:
\((x, y)\). The first number represents a position on the x-axis, and the second number represents a position on the y-axis. This concept is crucial for defining points in a coordinate system, where each pair corresponds to a unique location on a graph.
In the example given, the ordered pairs \({(4,5), (5,2), (1,6)}\) are used to define specific points in a two-dimensional space. Each pair indicates how far to move right or left along the x-axis and how far up or down along the y-axis to find the point. By understanding ordered pairs, you have a simple way to describe and plot points on a graph.

Domain and Range

The domain and range in mathematics describe the input and output values respectively in a relation or a function.
The domain is the set of all possible x-values (first elements in ordered pairs) that can be used in a function. These values are derived from the 'x' components of the ordered pairs. In our exercise example of the relation \({(4,5), (5,2), (1,6)}\), the domain is \({1, 4, 5}\), as these are all the x-coordinates from each pair.
The range is all the possible y-values (second elements in ordered pairs) that result from using the x-values in the function. From the ordered pairs given, the range is \({2, 5, 6}\), representing each distinct y-coordinate. Knowing the domain and range of a relation is essential for understanding the scope and limitation of a function's behavior.

Graphing Relations

Graphing relations involves representing ordered pairs visually on a coordinate plane. It provides insights into the structure and behavior of the relation. Each ordered pair like \((x, y)\), is plotted as a point where you move 'x' units horizontally and 'y' units vertically from the origin.
For the set \({(4,5), (5,2), (1,6)}\), you plot:

• Point \((4, 5)\) by moving 4 units to the right and 5 units up.
• Point \((5, 2)\) by moving 5 units to the right and 2 units up.
• Point \((1, 6)\) by moving 1 unit to the right and 6 units up.

Each point is a visual representation of the ordered pair and the entire graph shows the relationship among different points. Graphing helps in visualizing connections, patterns, or trends within the data.

Mathematical Tables

In mathematics, a table is used to organize data clearly and concisely. It represents information, like ordered pairs, in rows and columns so you can easily compare and analyze values.
For a relation like \({(4,5), (5,2), (1,6)}\), a table with columns for 'x' and 'y' allows you to see each pair's input and output at a glance. The table is set up as follows: | x | y ||---|---|| 4 | 5 || 5 | 2 || 1 | 6 |This format helps to quickly identify the domain and range by reading off the x-values and y-values respectively. Using mathematical tables is a straightforward method to organize and interpret relations, making complex data more digestible.