Sets and Set Operations

Set Notation and Operations

Sets are collections of distinct objects, often numbers. Common operations include union, intersection, and difference.

• Union (A ∪ B): The set of all elements in A or B.

• Intersection (A ∩ B): The set of elements common to both A and B.

• Difference (A \ B): The set of elements in A but not in B.

• Interval Notation: Used to describe subsets of real numbers, e.g., .

Example: If and , then .

Algebraic Operations and Simplification

Order of Operations and Exponents

Algebraic expressions can be simplified using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Exponent Rules:

• Fractional Exponents:

Example: Simplify .

Solving Equations and Inequalities

Solving Linear and Radical Equations

To solve equations, isolate the variable using inverse operations. For radical equations, eliminate the radical by raising both sides to the appropriate power.

• Linear Equation:

• Radical Equation:

Example: Solve by setting , so .

Solving Inequalities

Inequalities are solved similarly to equations, but solutions are often expressed in interval notation.

• Example: Solve .

Distance and Midpoint in the Coordinate Plane

Distance Formula

The distance between two points and is given by:

Midpoint Formula

The midpoint of a segment connecting and is:

Circles: Center and Radius

Standard Form of a Circle

The equation of a circle with center and radius is:

Example: For , complete the square to find center and radius.

Lines: Slope, Equation, and Graphing

Slope-Intercept Form

The equation of a line in slope-intercept form is:

Given two points, the slope is:

Parallel Lines: Same slope. Perpendicular Lines: Slopes are negative reciprocals.

Functions and Correspondence Diagrams

Definition of a Function

A function is a relation where each input has exactly one output.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

Example: Use correspondence diagrams to determine if a relation is a function.

Graphing Circles and Other Curves

Equation of a Circle

Standard form:

• Example:

Even and Odd Functions

Definitions

• Even Function: for all in the domain.

• Odd Function: for all in the domain.

• Neither: If neither condition is met.

Example:

Transformations of Functions

Shifts and Stretches

Transformations include vertical/horizontal shifts, stretches, and reflections.

• Vertical Shift: shifts up/down.

• Horizontal Shift: shifts right/left.

• Reflection: reflects over the x-axis.

Example:

Function Composition and Domains

Composition of Functions

The composition means .

• Domain of Composition: The set of such that is in the domain of and is in the domain of .

Example: If and , then .

Table: Properties of Functions

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