Overview
This lecture introduces functions in general mathematics, focusing on how to identify, represent, and interpret functions using ordered pairs, mapping diagrams, graphs, and equations.
Introduction to Functions
- Functions relate each element of a domain (input) to exactly one element of a range (output).
- Relations are sets of ordered pairs; a function is a specific type of relation.
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 pairs (1, -1), (2, -3), (0, 5), the domain is {-1, 0, 1, 2, 4}, the range is {-5, -4, -3, -1, 3, 5}.
Representations of Functions
- Functions can be represented as ordered pairs, tables, mapping diagrams, graphs, and equations.
- Ordered pairs: Check if each x-value points to only one y-value.
- Mapping diagrams: Each input must connect to only one output for a function.
Identifying Functions
- If any x-value (input) maps to more than one y-value (output), it is not a function.
- Use the vertical line test on a graph: if any vertical line crosses the graph more than once, it is not a function.
Functions in Graphs and Equations
- Linear and quadratic equations usually represent functions (e.g., y = 2x + 1, y = x² - 2x + 2).
- Equations like x² + y² = 1 do not represent functions, as one x-value yields two y-values.
- Rational or square root equations can be functions if each x produces only one y.
Practice and Examples
- Practice problems involve identifying if lists of ordered pairs, mapping diagrams, or graphs are functions.
- Apply the vertical line test and check for repeated x-values with different y-values.
Key Terms & Definitions
- Domain — set of all possible input values (x-values).
- Range — set of all possible output values (y-values).
- Function — a relation where each element of the domain maps to one element of the range.
- Ordered Pair — a pair (x, y) representing a relation between x and y.
- Mapping Diagram — visual representation of how each input is paired with an output.
- Vertical Line Test — a method to determine if a graph represents a function.
Action Items / Next Steps
- Complete the 10 practice questions given in the lecture.
- Review identifying functions using ordered pairs, mapping diagrams, graphs, and equations.
- Prepare for the next lesson on determining domain and range from equations.