Problem 26 - Prealgebra

Question

Express each relation as a table and as a graph. Then determine the domain and range. $$\\{(6,8),(2,9),(0,1)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{6, 2, 0\}\); Range: \(\{8, 9, 1\}\).

Step 1: Create a Table of Values

The given relation is a set of ordered pairs: \(\{(6,8), (2,9), (0,1)\}\).To express it as a table, list each ordered pair using two columns labeled 'x' and 'y'.| x | y ||---|---|| 6 | 8 || 2 | 9 || 0 | 1 |

Step 2: Plot the Points on a Graph

To represent the relation as a graph, plot each ordered pair as a point on a coordinate plane: - Plot the point (6, 8) - Plot the point (2, 9) - Plot the point (0, 1)

Step 3: Determine the Domain

The domain of a relation is the set of all first elements (x-coordinates) from the ordered pairs.For the given relation \(\{(6,8), (2,9), (0,1)\}\), the domain is \(\{6, 2, 0\}\).

Step 4: Determine the Range

The range of a relation is the set of all second elements (y-coordinates) from the ordered pairs.For the given relation \(\{(6,8), (2,9), (0,1)\}\), the range is \(\{8, 9, 1\}\).

Key Concepts

Domain and Range

In mathematics, the domain and range are crucial concepts when dealing with relations and functions. The **domain** refers to all the possible input values, often represented as the "x" values in ordered pairs, while the **range** is the set of possible output values, shown as the "y" values.

For example, consider the relation given by the set of ordered pairs \((6,8), (2,9), (0,1)\).

• The domain is composed of the first elements from each ordered pair: \(\{6, 2, 0\}\).
• The range includes the second elements: \(\{8, 9, 1\}\).

Understanding the domain helps us know which values the function or relation can accept, while the range shows us the possible outputs.

Ordered Pairs

An ordered pair is a fundamental concept in coordinate geometry, typically expressed as \((x, y)\). Each ordered pair contains two numbers, with the first number representing the **horizontal position** (often called x-coordinate) and the second number representing the **vertical position** (y-coordinate) on a coordinate plane.

In the ordered pairs \((6,8), (2,9), (0,1)\), each pair corresponds to a specific point in a two-dimensional space:

• Point \((6,8)\) has an x-coordinate of 6 and a y-coordinate of 8.
• Point \((2,9)\) has an x-coordinate of 2 and a y-coordinate of 9.
• Point \((0,1)\) has an x-coordinate of 0 and a y-coordinate of 1.

Ordered pairs allow us to precisely locate points on a graph or map, serving as the basic building blocks for plotting points on a coordinate plane.

Plotting Points

Plotting points is the process of marking locations on a coordinate plane using ordered pairs. This visual representation helps in understanding the structure and behavior of relations and functions.

Here's how to plot the points from the ordered pairs \((6,8), (2,9), (0,1)\):

• Start at the origin, where the x-axis and y-axis meet, which is \((0,0)\).
• For the point \((6,8)\), move 6 units right on the x-axis and 8 units up on the y-axis.
• For \((2,9)\), move 2 units right and 9 units up.
• For \((0,1)\), do not move horizontally but move 1 unit up vertically.

After plotting, these points clearly show the pattern and help determine further characteristics like the relationship between x and y values.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent mathematical relationships using ordered pairs. It consists of two perpendicular lines: the **x-axis**, which is horizontal, and the **y-axis**, which is vertical. Their intersection point is called the **origin** with coordinates \((0, 0)\).

Each point on this plane is identified uniquely by an ordered pair \((x, y)\). This makes it easy to visualize and analyze relationships between values. For example, the coordinate plane allows us to see:

• How changes in x affect y in a relation or function.
• The distribution and alignment of data points.
• Patterns such as trends or clusters.

The use of a coordinate plane is instrumental in not only plotting points but also in examining more complex mathematical ideas.