Relations and Functions | Algebra

Introduction

This tutorial provides a comprehensive overview of relations and functions in algebra. It will guide you through the essential concepts such as writing the domain and range of a relation, determining if a relation is a function, and using the vertical line test on graphs. By the end, you'll have a solid understanding of these foundational algebraic concepts.

Step 1: Understand Relations

• A relation is a set of ordered pairs.
• Example: The relation {(1, 2), (2, 3), (3, 4)} consists of three pairs where the first number is related to the second.

Step 2: Identify Domain and Range

• Domain: The set of all possible first elements (x-values) in a relation.
• Range: The set of all possible second elements (y-values) in a relation.

How to find domain and range:

1. List all unique x-values for the domain.
2. List all unique y-values for the range.

Example: For the relation {(1, 2), (2, 3), (3, 4)}:

• Domain: {1, 2, 3}
• Range: {2, 3, 4}

Step 3: Determine if a Relation is a Function

• A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
• To check if a relation is a function:
  • Look for repeated x-values. If any x-value has more than one corresponding y-value, it is not a function.

Example:

• Relation {(1, 2), (1, 3)} is not a function (1 maps to both 2 and 3).
• Relation {(1, 2), (2, 3)} is a function (1 maps to 2 and 2 maps to 3).

Step 4: Create a Mapping Diagram

• A mapping diagram visually represents the relation.

1. Write the domain values on one side and the range values on the other.
2. Draw arrows from each domain value to its corresponding range value.

Example: For the relation {(1, 2), (2, 3)}:

• Draw an arrow from 1 to 2 and another from 2 to 3.

Step 5: Construct a Function Table

• A function table lists pairs of inputs (x-values) and their corresponding outputs (y-values).

1. Create a table with two columns: one for x and one for y.
2. Fill in the pairs based on the relation.

Example: | x | y | |---|---| | 1 | 2 | | 2 | 3 |

Step 6: Use the Vertical Line Test

• The vertical line test is a method to determine if a graph represents a function.
• Procedure:
  1. Draw a vertical line through the graph at any point.
  2. If the vertical line intersects the graph at more than one point, it is not a function.

Conclusion

In this tutorial, you learned about relations and functions, including how to identify the domain and range, determine if a relation is a function, and visualize the relation using mapping diagrams and function tables. You also explored the vertical line test for graphs. Understanding these concepts is crucial for further study in algebra and related mathematical fields. Next, consider practicing with different sets of relations to reinforce your understanding.