Overview
This lecture introduces the concept of functions in mathematics, explains how to determine if a given set of pairs, mapping diagram, graph, or equation represents a function, and distinguishes between functions and non-functions.
Functions and Relations
- A relation is a set of ordered pairs.
- A function is a relation where each element of the domain maps to exactly one element in the range.
Domain and Range
- The domain is the set of first coordinates (x-values) in ordered pairs.
- The range is the set of second coordinates (y-values) in ordered pairs.
- Example: For (1, -1), (2, -3), (0, 5), the domain is {-1, 0, 1, 2, 4}; the range is {-5, -4, -3, -1, 3, 5}.
Ways to Represent Functions
- Functions can be represented using ordered pairs, tables, mapping diagrams, graphs, and equations.
Identifying Functions from Ordered Pairs
- A set of ordered pairs is a function if each x-value is paired with only one y-value (no repeated x-values with different y-values).
Mapping Diagrams
- In a mapping diagram, if an input (x-value) maps to more than one output (y-value), it is not a function.
- Only one arrow should come from each input to a single output for it to be a function.
Vertical Line Test for Graphs
- A graph represents a function if any vertical line intersects it at most once.
- If a vertical line intersects the graph more than once, it is not a function.
Functions from Equations
- Linear equations (e.g., y = 2x + 1) and quadratic equations (e.g., y = x² - 2x + 2) are functions.
- Equations like x² + y² = 1 are not functions because a single x can produce multiple y-values.
Practice Questions Overview
- Given sets of ordered pairs or mapping diagrams, identify if they are functions.
- Use the vertical line test for graphs to determine if they're functions.
- Decide if given equations describe a function or not.
Key Terms & Definitions
- Relation — a set of ordered pairs.
- Function — a relation where every x-value maps to exactly one y-value.
- Domain — set of all possible input values (x-values).
- Range — set of all possible output values (y-values).
- Mapping Diagram — a visual representation indicating how elements of the domain map to elements of the range.
- Vertical Line Test — a test to determine if a graph is a function; no vertical line should intersect more than once.
Action Items / Next Steps
- Review how to determine if an ordered pair, mapping diagram, graph, or equation is a function.
- Practice with additional sets of ordered pairs, mapping diagrams, graphs, and equations.
- Prepare for upcoming lessons on finding the domain and range from equations.