Functions and Their Properties

Definition of a Function

A function is a relation that assigns to each element in a set (the domain) exactly one element in another set (the range). Functions are fundamental in calculus, as they describe how one quantity changes with respect to another.

• Notation: denotes the value of the function at .

• Domain: The set of all possible input values () for which the function is defined.

• Range: The set of all possible output values ().

Types of Functions

• Linear Function:

• Quadratic Function:

• Polynomial Function:

• Rational Function: where

• Exponential Function:

• Logarithmic Function:

• Trigonometric Functions:

Function Operations

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

• Composition:

Graphs of Functions

The graph of a function is a visual representation of the set of ordered pairs . Key features include intercepts, asymptotes, and intervals of increase/decrease.

• Intercepts: Points where the graph crosses the axes.

• Asymptotes: Lines that the graph approaches but never touches.

• Vertex (for quadratics): The highest or lowest point of a parabola.

Example: Quadratic Function

The standard form is . The vertex is at .

Exponent Rules

Multiplying Powers with the Same Base

• General Rule:

• Example:

Dividing Powers with the Same Base

• General Rule:

• Example:

Finding a Power of a Power

• General Rule:

• Example:

Negative Exponents

• General Rule:

• Example:

Zero as an Exponent

• General Rule:

Algebraic Properties and Operations

Basic Properties

• Commutative Property: ,

• Associative Property: ,

• Distributive Property:

Absolute Value

• Definition: is the distance from to 0 on the number line.

• Properties: ,

Complex Numbers

• Definition: , where

• Conjugate:

Factoring and Solving Equations

• Quadratic Formula:

• Factoring:

Logarithmic and Exponential Functions

Logarithm Properties

• Product Rule:

• Quotient Rule:

• Power Rule:

Exponential Equations

• General Form:

• Inverse:

Trigonometric Functions and Identities

Definition of Trig Functions

• Sine:

• Cosine:

• Tangent:

Unit Circle

• On the unit circle, ,

• Common angles: , , , , , etc.

Key Trigonometric Identities

• Pythagorean Identity:

• Reciprocal Identities: , ,

• Sum and Difference Formulas:

Limits and Continuity

Definition of a Limit

The limit of a function as approaches is the value that gets closer to as gets closer to . Limits are foundational for calculus, especially in defining derivatives and integrals.

• Notation:

• Limit at Infinity:

• One-sided Limits: (from the left), (from the right)

Properties of Limits

• Sum Rule:

• Product Rule:

• Quotient Rule: ,

Evaluating Limits

• Direct substitution: If is continuous at , then

• Factoring and simplifying rational expressions

• Using conjugates for limits involving square roots

Continuity

• A function is continuous at if

• Discontinuities occur at points where the function is not defined or the limit does not exist.

Intermediate Value Theorem

If is continuous on and is any number between and , then there exists in such that .

Common Algebraic Errors

Misapplication of Properties

• Incorrectly distributing exponents:

• Incorrectly canceling terms in fractions

• Confusing rules for logarithms and exponents

Table: Common Algebraic Errors

Summary

These foundational concepts in algebra, functions, exponents, trigonometry, and limits are essential for success in calculus. Mastery of these topics enables students to understand derivatives, integrals, and advanced calculus applications.

Additional info: Some context and explanations have been expanded for clarity and completeness.