Overview
This lecture introduces the concept of functions in mathematics, emphasizing the rule-based relationship between inputs and outputs and providing examples and non-examples.
Definition and Properties of Functions
- A function is a relationship between two sets where each input matches exactly one output.
- The set of all possible inputs is called the domain.
- The set of all possible outputs is known as the codomain.
- It is acceptable for different inputs to share the same output.
- A function cannot have one input correspond to multiple outputs.
Ways to Specify Domain and Codomain
- Domains and codomains can be described using interval notation, inequalities, or plain language (e.g., “all real numbers”).
- Functions in calculus typically have numerical domains and codomains.
Examples of Functions
- Birthday Month Rule: Inputs are people, outputs are their unique birthday months; each person has one birthday month.
- Depth Rule: Inputs are times, outputs are water depths in a specified tank; one depth per time.
- Sine Function: Inputs are angles, outputs are y-values on a unit circle; one y-value per angle.
Non-Examples of Functions
- Pet Rule: Inputs are people, outputs are pets; a person may have multiple pets, violating the one-output rule.
- Elevation Rule: Inputs are elevations, outputs are times reached; one elevation can occur at multiple times.
- Square Root Rule: Inputs are numbers, outputs are square roots; each number has both a positive and negative root, unless restricted to the positive root.
Key Terms & Definitions
- Function — A rule assigning exactly one output to each input from a set.
- Domain — The set of all possible input values for a function.
- Codomain — The set of all potential output values for a function.
Action Items / Next Steps
- Reflect and list examples and non-examples of functions relevant to your experiences.