Overview
The lecture introduces the concept of functions in mathematics, focusing on how to identify and represent them using ordered pairs, mapping diagrams, graphs, and equations.
Domain and Range
- The domain is the set of all first coordinates (x-values) in a set of ordered pairs.
- The range is the set of all second coordinates (y-values) in a set of ordered pairs.
- To find the domain and range, list all x-values and y-values from the pairs, respectively.
Relations and Functions
- A relation is a set of ordered pairs.
- A function is a relation where each element of the domain corresponds to exactly one element of the range.
- If an x-value maps to more than one y-value, it is not a function.
Representing Functions
- Functions can be represented by: ordered pairs, tables of values, mapping diagrams, graphs, and equations.
- In mapping diagrams, each input should have only one output for the relation to be a function.
Identifying Functions
- For ordered pairs: check if each x-value is paired with only one y-value.
- For mapping diagrams: each input arrow should lead to only one output.
- For graphs: use the vertical line test; if any vertical line crosses the graph more than once, it is not a function.
- For equations: if solving for y gives more than one value for an x, it is not a function (e.g., circles).
Worked Examples
- Ordered pairs: sets where some x-values repeat with different y-values are not functions.
- Mapping diagrams: multiple arrows from one input to different outputs means not a function.
- Graphs: failing the vertical line test means not a function.
- Equations: linear and quadratic equations (like (y = 2x + 1) or (y = x^2-2x+2)) are functions; equations like (x^2 + y^2 = 1) are not.
Key Terms & Definitions
- Domain — set of all possible input values (x-values).
- Range — set of all possible output values (y-values).
- Relation — set of ordered pairs (x, y).
- Function — a relation where each domain element maps to exactly one range element.
- Vertical Line Test — a test to determine if a graph represents a function by checking if a vertical line crosses the graph more than once.
Action Items / Next Steps
- Practice identifying functions using ordered pairs, mapping diagrams, graphs, and equations.
- Complete the ten provided practice questions to reinforce understanding.
- Review next lesson: Determining domain and range from equations.