Problem 10 - Prealgebra

Question

Express each relation as a table and as a graph. Then determine the domain and range. $$\\{(1,6),(6,4),(0,2),(3,1)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{0, 1, 3, 6\}\), Range: \(\{1, 2, 4, 6\}\).

Step 1: Create a Table

List each ordered pair from the set \(\{(1,6),(6,4),(0,2),(3,1)\}\) into a two-column table. The first column will represent the \(x\)-values, and the second column will represent the \(y\)-values.| \(x\) | \(y\) ||---|---|| 1 | 6 || 6 | 4 || 0 | 2 || 3 | 1 |

Step 2: Plot the Graph

Create a Cartesian coordinate system and plot each ordered pair \((x, y)\). These points are: - \((1, 6)\)- \((6, 4)\)- \((0, 2)\)- \((3, 1)\)Mark each point clearly on the graph.

Step 3: Determine the Domain

The domain of a relation is the set of all possible \(x\)-values. From the ordered pairs \((1,6), (6,4), (0,2), (3,1)\), extract the \(x\)-values: - \(1\)- \(6\)- \(0\)- \(3\)Thus, the domain is \(\{0, 1, 3, 6\}\).

Step 4: Determine the Range

The range of a relation is the set of all possible \(y\)-values. From the ordered pairs \((1,6), (6,4), (0,2), (3,1)\), extract the \(y\)-values:- \(6\)- \(4\)- \(2\)- \(1\)Thus, the range is \(\{1, 2, 4, 6\}\).

Key Concepts

Domain and Range

Understanding the concepts of domain and range is crucial when working with relations in mathematics. These terms help us to outline the scope of a relation and identify the values involved. The **domain** refers to all the possible input values or the values on the horizontal axis, typically represented by the variable \( x \). In our case, for the given set of ordered pairs \( \{(1,6),(6,4),(0,2),(3,1)\} \), we extract the \( x \)-values: \( 0, 1, 3, \) and \( 6 \). This makes up the domain, and we can simply list it as \( \{0, 1, 3, 6\} \).

Similarly, the **range** involves all the possible output values or the values on the vertical axis, denoted typically by the variable \( y \). From the same set of ordered pairs, we identify the \( y \)-values: \( 6, 4, 2, \) and \( 1 \). Therefore, the range is \( \{1, 2, 4, 6\} \). Recognizing the domain and range helps us understand the extent and limits of the relationship between pairs in the relation.

Ordered Pairs

An ordered pair is a fundamental concept in mathematics that represents a pair of elements in a specific order. Ordered pairs are usually expressed in the form \((x, y)\), where \( x \) represents the first element, and \( y \) represents the second element. In the context of relations, ordered pairs serve to establish a relationship between two values.

For the given set of ordered pairs \( \{(1,6),(6,4),(0,2),(3,1)\} \), each pair communicates a link between \( x \) and \( y \). For example, the pair \((1, 6)\) indicates a relation where the input \( x = 1 \) leads to the output \( y = 6 \). Arranging these pairs systematically helps in identifying patterns, connections, and relationships that exist within the data set. They are an integral part of describing the graph or table of the relationship.

Graphing Coordinates

Graphing coordinates involves plotting ordered pairs on a Cartesian coordinate system. This provides a visual representation of a relation. When graphing, each ordered pair \((x, y)\) is represented as a point on a plane, where the \( x \)-coordinate determines the horizontal position and the \( y \)-coordinate determines the vertical position.

To plot our set of ordered pairs \( \{(1,6),(6,4),(0,2),(3,1)\} \), we begin by drawing a Cartesian plane with an \( x \)-axis (horizontal) and \( y \)-axis (vertical). We then plot each pair:

• Place a mark at \((1, 6)\) which means 1 unit along the \( x \)-axis and 6 units up the \( y \)-axis.
• Similarly, plot \((6, 4)\), \((0, 2)\), and \((3, 1)\).

Graphing allows us to see the relationship and behavior of points visually. It highlights confirmations of patterns, such as trends, clusters, or outliers, which might not be immediately evident from raw data or tables.