Overview
This lecture introduces the fundamental concept of functions in mathematics, covering their definition, notation, evaluation, and related key concepts such as domain, range, and how functions differ from relations.
Introduction to Functions
- A function is a rule that assigns each input exactly one output.
- The notation f(x) is read as "f of x" and represents a function named f with input x.
- An example function: f(x) = 3x² + 1.
Function Notation and Examples
- Functions can have different names, such as g(x) or h(t), depending on the variable used.
- f(x) and y are interchangeable; f(x) = y describes the output value.
Domain and Range
- The domain is the set of all possible input values (usually associated with x).
- The range is the set of all possible output values (usually associated with y).
Evaluating Functions
- To evaluate a function at a certain value, substitute the input for x in the rule.
- Example: To find f(2) for f(x) = 3x² + 1, calculate 3 × (2)² + 1 = 13.
- The ordered pair (input, output) gives a point (e.g., (2, 13)) on the function's graph.
Representing Functions
- Functions can be represented as equations, tables of values, or graphs.
- The graph of f(x) = 3x² + 1 is a parabola.
Relations vs. Functions
- A relation is any set of ordered pairs; a function is a specific type of relation where each input has only one output.
- All functions are relations, but not all relations are functions.
Additional Function Topics
- Determining whether a relation is a function is a key skill.
- Functions can be combined or manipulated (e.g., added, multiplied, composed).
- The concept of a function inverse is also important.
Key Terms & Definitions
- Function — A rule that assigns each input exactly one output.
- f(x) — Function notation, meaning the output of f for input x.
- Domain — The set of all input values for a function.
- Range — The set of all output values for a function.
- Relation — Any set of ordered pairs (input, output).
- Inverse Function — A function that reverses the effect of the original function.
Action Items / Next Steps
- Practice evaluating functions for various inputs.
- Learn how to determine if a relation is a function.
- Study the concepts of function operations and inverse functions.
- Suggested: Review domain and range in your math textbook or algebra course.