Problem 26 - Algebra 2

Question

Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function and state whether it is discrete or continuous. $$ \\{(3,4),(4,3),(6,5),(5,6)\\} $$

Step-by-Step Solution

Answer

Domain: \(\{3, 4, 5, 6\}\); Range: \(\{3, 4, 5, 6\}\); The relation is a discrete function.

Step 1: Plot the Points

To begin analyzing the given relation \(\{(3,4),(4,3),(6,5),(5,6)\}\), first, we plot these points on a Cartesian coordinate system. Each pair \((x, y)\) represents a point, where the first number is the x-coordinate, and the second number is the y-coordinate.

Step 2: Find the Domain

The domain refers to the set of all possible x-values that the relation includes. From the points (3, 4), (4, 3), (6, 5), and (5, 6), the x-values are 3, 4, 6, and 5. Therefore, the domain is \(\{3, 4, 5, 6\}\).

Step 3: Find the Range

The range is the set of all possible y-values from the relation. Looking at the given points: (3, 4), (4, 3), (6, 5), and (5, 6), the y-values are 4, 3, 5, and 6. Thus, the range is \(\{3, 4, 5, 6\}\).

Step 4: Determine if it is a Function

For a relation to be a function, each x-value must be associated with exactly one y-value. Reviewing the points, each x-value corresponds to only one y-value, confirming that this relation is indeed a function.

Step 5: Determine if it is Discrete or Continuous

This relation is made up of individual distinct points, not connected in a continuous curve or line. Therefore, it is a discrete relation.

Key Concepts

Domain

The domain in the context of a function or relation is an essential concept to understand. It refers to all possible input values (x-values) that a function can accept. For the relation \(\{(3,4),(4,3),(6,5),(5,6)\}\), the domain is simply the set of all unique x-coordinates:

• From the point (3, 4), the x-value is 3.
• From the point (4, 3), the x-value is 4.
• From the point (6, 5), the x-value is 6.
• From the point (5, 6), the x-value is 5.

Putting these together, the domain is \(\{3, 4, 5, 6\}\). It is crucial to list each value only once, even if it appears in multiple points. In plotting functions or relations, determining the domain helps you understand the scope within which the function acts.

Range

The range of a function or relation complements the domain by identifying all possible output values (y-values). Let's look at the given relation \(\{(3,4),(4,3),(6,5),(5,6)\}\) to find out which y-values are included:

• From the point (3, 4), the y-value is 4.
• From the point (4, 3), the y-value is 3.
• From the point (6, 5), the y-value is 5.
• From the point (5, 6), the y-value is 6.

The range is \(\{3, 4, 5, 6\}\). Again, similar to the domain, each value should be listed only once. Understanding the range is crucial as it tells you the set of possible outcomes or results produced by the relation. This can help predict or understand behaviors within the relationship being graphed or analyzed.

Discrete and continuous relations

When discussing relations and functions, it's important to understand whether a relation is discrete or continuous. These terms describe the nature of the connection between the domain and range.

Discrete Relations

A discrete relation consists of distinct, separate values. In our example, the relation \(\{(3,4),(4,3),(6,5),(5,6)\}\) is discrete because each point is isolated with no points in between. You can count the number of points, which are not connected by a smooth line or curve. This is typical in sets like lists of specific data points.

Continuous Relations

In contrast, a continuous relation has points that are connected smoothly through lines or curves, without any gaps. This means values can be anything within a range, including fractions or decimals.Understanding whether a relation is discrete or continuous affects how the data is interpreted and how we might choose to represent it graphically. Discrete relations are represented by individual points, while continuous relations can be drawn as lines or curves.