Lecture Notes: Understanding Functions

Overview

This lecture aims to provide a comprehensive understanding of functions, covering several types and properties including:

• Relations and functions
• Inverse functions
• Composite functions
• Quadratic functions
• Domain and range
• Piecewise functions
• Odd and even functions
• Polynomials
• Rational functions
• Modulus functions
• Radical functions

Key Concepts

Relations and Functions

• Relation: A set of pairs of input (x) and output (y) values.
• Domain: Set of all possible x-values (inputs) in a function.
• Range: Set of all possible y-values (outputs) in a function.
• For a relation to be a function, each input must have exactly one output.

Types of Functions

• One-to-One Function: Each element of the domain maps to a unique element in the range.
• Many-to-One Function: Multiple elements from the domain map to the same element in the range.
• One-to-Many Function and Many-to-Many Function: Not considered functions because a single input maps to multiple outputs.

Inverse Functions

• The inverse function f^(-1)(x) reverses the action of f(x), such that f(f^(-1)(x)) = x.
• Finding the inverse involves solving the equation y = f(x) for x and then swapping x and y.

Composite Functions

• A composite function f(g(x)) involves applying one function to the result of another: substitute g(x) into f(x).

Domain and Range

• Domain: All possible x-values for which the function is defined.
• Range: All possible y-values that the function can output.
• Specific rules apply for functions involving fractions and roots (e.g., denominators cannot be zero).

Piecewise Functions

• Functions that have different expressions based on the domain subset.
• Each piece has its own domain and is evaluated separately.

Odd and Even Functions

• Even Function: f(-x) = f(x) for all x. Symmetric about the y-axis.
• Odd Function: f(-x) = -f(x) for all x. Symmetric about the origin.

Quadratic Functions

• In standard form ax^2 + bx + c.
• Completing the Square: A method to find vertex form and identify the vertex (turning point).

Sketching Graphs

• Consider the vertex, axis of symmetry, and intercepts.
• For functions with modulus, reflect portions of the graph across the x-axis where necessary.

Examples and Applications

• Various examples were provided to illustrate how to determine if a relation is a function, find the inverse of a function, and solve for composite functions.
• Graphing techniques included identifying key points, symmetry, and intercepts.
• The lecture concludes with exercises on the remainder theorem and computing square methods involving modulus functions.

Summary

Understanding functions involves recognizing their different forms, properties, and transformations. Mastering these concepts allows for accurate graphing and solving of function-related problems.