Overview
This lecture introduces the topic of functions of several variables, focusing on definitions, representations, and key characteristics.
Functions: Basic Concepts
- A function mathematically relates each input (independent variable) to a single output (dependent variable).
- The main characteristics of a function are: a rule, a set of inputs (domain), and each input has exactly one output (range).
Functions of One Variable
- A function of one variable is commonly written as f(x), where x is the independent variable and f(x) is the dependent variable.
- This notation clearly identifies the input and output of the function.
Functions of Two Variables
- A function of two variables is written as z = f(x, y), where z is the dependent variable and x, y are independent variables.
- The form z = f(x, y) shows how the output depends on two inputs.
Functions of Several Variables
- A function of several variables is expressed as f(x₁, x₂, ..., xₙ), where n is any natural number.
- Subscripts (x₁, x₂, ..., xₙ) distinguish between different independent variables.
- Using subscripts provides a consistent way to denote many variables without using multiple letters.
Key Terms & Definitions
- Function — A rule that assigns each element from a set (domain) to exactly one element in another set (range).
- Independent Variable — The input(s) of a function; values that can be varied freely.
- Dependent Variable — The output of a function, determined by the value(s) of the independent variable(s).
- Subscript Notation — Using indices (e.g., x₁, x₂) to distinguish multiple variables in functions of several variables.
Action Items / Next Steps
- Reflect and write your own definitions and representations of functions with one, two, or several variables.
- Note any questions or comments to discuss in the group session.