Lecture Notes: Relations and Functions

Key Concepts

• Relations and Functions: A relation associates elements of one set with another set. A function is a special type of relation where each element of the first set is associated with a unique element of the second set.
• Linear Functions: A function where the rate of change is constant, and its graph is a straight line.

Vocabulary

• Relation: An association between two sets.
• Function: A type of relation with unique associations.
• Domain: The set of all possible input values.
• Range: The set of all possible output values.
• Rate of Change: The change in the output value relative to the change in input value.
• Vertical Intercept: The point where the graph intersects the vertical axis.
• Horizontal Intercept: The point where the graph intersects the horizontal axis.

Representing Relations

• Relations can be represented via tables, graphs, and arrow diagrams.
• Example: Associating fruits with their colors using ordered pairs or a table.

Tables and Graphs

• Linear Relations: Recognized by a constant change in the independent and dependent variables.
• Graphing: A linear relation will produce a straight line.

Identifying Functions

• Use the Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Examples

• Set of Ordered Pairs: Demonstrating both linear and non-linear relations.
• Graphs: Determining functions through visual representation and intercepts.

Interpreting Graphs

• Understanding real-world applications, such as fuel consumption or distance traveled over time.

Exercises

• Practicing the representation and analysis of relations and functions through various exercises involving tables, graphs, and equations.

Summary

• Reinforcement of the concept of linear functions and their properties such as domain, range, and intercepts.
• Application of the concepts to solve practical problems and understanding different ways to represent relations and identify functions.

These notes cover the fundamental concepts of relations and functions, focusing on how to identify, represent, and analyze them through mathematical models and real-world examples. It is important to practice these concepts through exercises to better understand their applications.