Problem 28 - Prealgebra

Question

Express each relation as a table and as a graph. Then determine the domain and range. $$\\{(2,4),(1,3),(5,6),(1,1)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{1, 2, 5\}\); Range: \(\{1, 3, 4, 6\}\).

Step 1: Expressing the Relation in a Table

To create a table from the given set of ordered pairs \(\{(2,4), (1,3), (5,6), (1,1)\}\), list each pair in separate rows with two columns representing the 'x' and 'y' values respectively. The table will look like this: \[\begin{array}{c|c}\text{x} & \text{y} \\hline2 & 4 \1 & 3 \5 & 6 \1 & 1 \\end{array}\]

Step 2: Plotting the Relation on a Graph

Graph each pair as a point on a two-dimensional plane, where the first element of the pair is the 'x-coordinate' and the second element is the 'y-coordinate'. 1. Plot the point (2,4). 2. Plot the point (1,3). 3. Plot the point (5,6). 4. Plot the point (1,1). Notice that (1,3) and (1,1) have the same x-coordinate.

Step 3: Determining the Domain

The domain of a relation is the set of all possible x-values from the ordered pairs. For the relation \(\{(2,4), (1,3), (5,6), (1,1)\}\), the x-values are \(2, 1, 5\). As there are repeated instances of the value 1, the domain is \(\{1, 2, 5\}\).

Step 4: Determining the Range

The range of a relation is the set of all possible y-values from the ordered pairs. For the relation \(\{(2,4), (1,3), (5,6), (1,1)\}\), the y-values are \(4, 3, 6, 1\). Thus, the range is \(\{1, 3, 4, 6\}\).

Key Concepts

Domain and Range

When dealing with relations in mathematics, understanding the domain and range is essential. The **domain** of a relation is the set of all possible x-values from the ordered pairs. Think of it as all the "input" values. For example, in the relation \(\{(2,4), (1,3), (5,6), (1,1)\}\), the domain includes the x-values: \(2, 1, 5\). It's important to note that while \(1\) appears twice, it only gets counted once in the domain set: \(\{1, 2, 5\}\).

The **range** is similar but focuses on the y-values. It is the set of all possible y-values or "output" values. In our example, examining the relation \(\{(2,4), (1,3), (5,6), (1,1)\}\), we find the y-values are \(4, 3, 6, 1\). Unlike the domain, each y-value is unique for this relation, so our range is \(\{1, 3, 4, 6\}\).

In summary:

• Domain: Set of all x-values.
• Range: Set of all y-values.
• Values in both domain and range are listed once, even if they repeat in the pairs.

Ordered Pairs

Ordered pairs are fundamental in expressing mathematical relations. Each ordered pair contains two components: the first element is the x-coordinate, and the second element is the y-coordinate. They are typically represented in the form \((x, y)\) and are used to show relationships between two quantities.

In the given relation \(\{(2,4), (1,3), (5,6), (1,1)\}\), each pair represents a point. For example, the pair \((2,4)\) means when the x-value is \(2\), the corresponding y-value is \(4\).

You can think of ordered pairs as directions:

• The first number indicates how far along the x-axis the point is."
• The second number indicates how far up or down the y-axis it is."

This dual property makes ordered pairs extremely useful for visualizing data on a graph and in reflecting real-world relationships.

Each unordered pair is unique in its position on the Cartesian plane, making it easy to identify, compare, and analyze relationships.

Graphing Relations

Graphing relations involves plotting the ordered pairs on a two-dimensional coordinate plane. Let’s visualize this with our example set \(\{(2,4), (1,3), (5,6), (1,1)\}\).

Each ordered pair is plotted as a point, where the first number is the position on the x-axis, and the second is on the y-axis. Here's how you can plot these:

• Start by locating the x-value on the x-axis.
• Move vertically to where you find the corresponding y-value on the y-axis.
• Mark this intersection as a distinct point - this is your ordered pair.

For the pair \((2,4)\), you would find the number 2 on the x-axis, then move up to the point that aligns with 4 on the y-axis, and place a point there. Repeat this for the rest of the pairs.

Graphing gives a visual representation of the relationship defined by such pairs. You can see:

If any x-values share the same y-value or vice versa.

How the pairings are distributed across the plane.

This visual aspect is beneficial for better understanding and identifying the characteristics of the relation, such as patterns and trends.