Overview
This lesson introduces the concept of functions in general mathematics, focusing on their definitions, identification, and representation through mapping, sets, graphs, and real-life examples.
Objectives of the Lesson
- Determine the difference between functions and relations.
- Illustrate functions using mapping diagrams, sets, and graphs.
- Represent real-life situations using functions.
Key Concepts: Relations and Functions
- A relation relates values from a set called the domain to another set called the range.
- An ordered pair is a pair of objects in a specific order, written as (x, y).
- The domain is the set of all input (x) values; the range is the set of all output (y) values.
- A set is a collection of distinct objects with a common characteristic.
Definition of a Function
- A function is a special type of relation where each x value in the domain is paired with exactly one y value in the range.
- If any x value is paired with more than one y value, it is not a function.
Illustrating Functions
Mapping Diagrams
- If every element in the domain maps to only one element in the range, it is a function.
- It's acceptable for different x values to map to the same y value.
Sets (Roster Notation)
- When no x value is repeated in the set of ordered pairs, the set is a function.
- If an x value appears more than once with different y values, it is not a function.
Graphs
- Use the Vertical Line Test (VLT): a graph represents a function if any vertical line intersects the graph at only one point.
- If a vertical line touches the graph more than once, it is not a function.
Real-Life Examples of Functions
- Circumference of a circle as a function of its diameter (C = dฯ) or radius (C = 2ฯr).
- Length of a personโs shadow as a function of their height.
- Location of a car as a function of time while driving.
Key Terms & Definitions
- Relation โ a rule that links elements from one set (domain) to another set (range).
- Function โ a relation where each input (x) has exactly one output (y).
- Ordered Pair โ two elements written as (x, y), showing the input and output.
- Domain โ the set of all possible input values (x).
- Range โ the set of all possible output values (y).
- Vertical Line Test (VLT) โ a method to determine if a graph represents a function by checking vertical line intersections.
Action Items / Next Steps
- Complete the 10-item assessment provided and submit your answers to your teacher as scheduled.
- Review function definitions and practice identifying functions through mapping, sets, and graphs.