Problem 27 - Prealgebra

Question

Express each relation as a table and as a graph. Then determine the domain and range. $$\\{(7,0),(3,2),(4,4),(5,1)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{7, 3, 4, 5\}\); Range: \(\{0, 2, 4, 1\}\).

Step 1: Express the Relation as a Table

To express the relation as a table, list each pair of values in separate rows with labeled columns for the x-values and y-values. | x | y | |---|---| | 7 | 0 | | 3 | 2 | | 4 | 4 | | 5 | 1 |

Step 2: Plot the Relation on a Graph

To graph the relation, plot each pair of coordinates on a Cartesian plane. Each pair \((x, y)\) corresponds to a point on the graph.1. Plot the point (7, 0) on the plane.2. Plot the point (3, 2) on the plane.3. Plot the point (4, 4) on the plane.4. Plot the point (5, 1) on the plane.Connect these dots only if the context of the task suggests the points are connected.

Step 3: Determine the Domain

The domain of a relation is the set of all x-values from the pairs. Here, list the x-values obtained from the relation.Domain: \( \{7, 3, 4, 5\} \).

Step 4: Determine the Range

The range of a relation is the set of all y-values from the pairs. Here, list the y-values obtained from the relation.Range: \( \{0, 2, 4, 1\} \).

Key Concepts

Table Representation

When expressing a relation as a table, we can easily organize the information and make it more understandable. A table contains two columns: one for the x-values and one for the y-values. Each row in the table corresponds to a pair of values from the relation. For example, if our pairs of values are \((7, 0), (3, 2), (4, 4), (5, 1)\), our table will look something like this:
- The first column lists all x-values: 7, 3, 4, 5.- The second column lists all corresponding y-values: 0, 2, 4, 1.
This tabular representation helps in quickly visualizing and organizing data. It's especially beneficial for seeing directly the connection between each x-value and its corresponding y-value.

Graphing Coordinates

Graphing coordinates involves plotting pairs of points on a Cartesian plane. This is a visual way to represent the relation between x and y-values. Each pair \((x, y)\) is a point that you can plot. Here’s how to plot the points from our previous example:

• Locate the point \((7, 0)\) by moving 7 units along the x-axis and 0 units up or down the y-axis.
• Locate the point \((3, 2)\) by moving 3 units along the x-axis and 2 units up the y-axis.
• Repeat this process to plot \((4, 4)\) and \((5, 1)\).

Connecting the dots is optional and depends on whether the points represent a continuous function or just distinct points. By graphing these points, you get a clear visual representation of the relation.

Domain and Range

The domain and range of a relation give us crucial information about the set of possible input and output values:- **Domain**: The domain consists of all the x-values from our set of points. For the example \((7, 0), (3, 2), (4, 4), (5, 1)\), the domain is \( \{7, 3, 4, 5\} \).- **Range**: The range includes all the y-values from the same set of points. Thus, the range is \( \{0, 2, 4, 1\} \).
Understanding the domain and range is essential to knowing which values x and y can take. This determines how the function or relation behaves.

X-Values

X-values represent the input or independent values in a set of ordered pairs, appearing in the form \((x, y)\). Each x-value corresponds to an independent choice or input. In our working example, x-values are 7, 3, 4, and 5. They are often used to find the corresponding function value (y-value) by evaluating a function associated with the relationship.
The x-values are what you use to navigate horizontally on a graph. They determine the position along the x-axis, illustrating how the input changes over the sample space. In real-world problems, they often represent time, quantity, distance, etc.

Y-Values

Y-values are the dependent values in a relation. Each y-value is determined by the respective x-value, forming pairs as \((x, y)\). In our example, the y-values are 0, 2, 4, and 1. They reflect the effect or outcome when the corresponding x-value is used in a function.
When plotting on a graph, y-values indicate the vertical position—the greater the y-value, the higher the point is on the y-axis. Understanding y-values in context helps us see the result of changes in x-values, and how one interacts with another in the relation or function.