Problem 29 - Prealgebra

Question

Express each relation as a table and as a graph. Then determine the domain and range. $$\\{(0,1),(0,3),(0,5),(2,0)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{0, 2\}\), Range: \(\{0, 1, 3, 5\}\).

Step 1: List the Ordered Pairs

The given relation is a set of ordered pairs: \((0,1)\), \((0,3)\), \((0,5)\), and \((2,0)\).

Step 2: Create a Table of the Relation

Organize the ordered pairs into a table with two columns, where the first column represents the input values (x-coordinates) and the second column represents the output values (y-coordinates):\[\begin{array}{c|c}x & y \\hline0 & 1 \0 & 3 \0 & 5 \2 & 0 \\end{array}\]

Step 3: Plot the Relation on a Graph

Plot each ordered pair on a Cartesian coordinate plane. Place a point for each pair where the x-coordinate determines the horizontal position and the y-coordinate determines the vertical position. This will create four points: \((0,1)\), \((0,3)\), \((0,5)\), and \((2,0)\).

Step 4: Determine the Domain

The domain of a relation is the set of all possible x-values from the ordered pairs. From the given pairs, the domain is \(\{0, 2\}\).

Step 5: Determine the Range

The range of a relation is the set of all possible y-values from the ordered pairs. From the given pairs, the range is \(\{0, 1, 3, 5\}\).

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is simply a pair of numbers used to locate a point on the coordinate plane. The ordered pair format is (x, y), where:

• The first number (x) is the x-coordinate, indicating the horizontal position.
• The second number (y) is the y-coordinate, showing the vertical position.

This pair ensures that each point on a plane can be described uniquely, as swapping the numbers changes the location they describe. For instance, (2, 3) is not the same as (3, 2). Understanding ordered pairs is crucial, as they are the foundation for graphing on a coordinate plane.

Domain and Range

The concept of domain and range comes into play when dealing with relations and functions in mathematics. For a set of ordered pairs, the domain refers to all the unique x-values that appear, while the range includes all the y-values.

• Domain: It is the set of all input values, or the first elements of each ordered pair. In our example, the domain is \(\{0, 2\}\), emphasizing that 0 repeats does not alter the set's uniqueness.
• Range: This set consists of all output values, or the second elements of the ordered pairs. The range here is \(\{0, 1, 3, 5\}\), indicating the various heights on the plane.

Understanding domain and range helps in identifying which values a relation can legitimately take, thereby assisting in analyzing its behavior more clearly.

Graphing on a Coordinate Plane

Visualizing ordered pairs on a graph provides a concrete representation of relations. The coordinate plane, also known as the Cartesian plane, consists of two perpendicular lines called axes:

• x-axis: The horizontal line where values increase rightwards and decrease leftwards.
• y-axis: The vertical line where values increase upwards and decrease downwards.

Points are plotted based on their ordered pairs by first finding the x-coordinate along the x-axis, then moving vertically to the y-coordinate. For instance, to plot (0, 1), start at the origin (0, 0) and move up one unit along the y-axis. Graphing these points allows for an easy visualization of relations and patterns, making it a critical skill for analyzing mathematical data. Moreover, each point tells a story about how a set of inputs relates to a set of outputs within a given context.