Number Sets and Classification

Types of Numbers

Understanding the classification of numbers is foundational in precalculus. Numbers are grouped into several sets based on their properties.

• Natural Numbers: Counting numbers starting from 1. Examples: 1, 2, 3, ...

• Whole Numbers: Natural numbers plus zero. Examples: 0, 1, 2, 3, ...

• Integers: Whole numbers and their negatives. Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Rational Numbers: Numbers that can be written as a fraction of two integers. Examples: , ,

• Irrational Numbers: Numbers that cannot be written as a fraction of two integers. Examples: , ,

• Real Numbers: All rational and irrational numbers.

Example: Classify the numbers -1.5, , -0.42, -0.4, 1.4, into the above categories.

Graphing Points and Intervals

Plotting Points and Quadrants

Points in the plane are represented as (x, y). The plane is divided into four quadrants.

• Quadrant I: (+, +)

• Quadrant II: (-, +)

• Quadrant III: (-, -)

• Quadrant IV: (+, -)

Example: Plot the points (2, 4) and (0, -3) and state their quadrants.

Interval Notation

Intervals describe sets of numbers between two endpoints.

Example: Express in interval notation.

Functions and Their Properties

Definition of a Function

A function is a relation in which each input (x-value) has exactly one output (y-value).

• Vertical Line Test: If a vertical line intersects a graph more than once, the graph is not a function.

Example: Determine if the relation {(1,5), (3,2), (4,9), (7,6)} is a function.

Domain and Range

• Domain: The set of all possible input values (x-values).

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

Example: Find the domain and range of .

Linear Equations and Slope

Slope of a Line

The slope measures the steepness of a line. For points and :

• Positive slope: Line rises left to right.

• Negative slope: Line falls left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Slope-Intercept Form

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

• m: slope

• b: y-intercept

Example: Find the slope and y-intercept of .

Parallel and Perpendicular Lines

• Parallel lines: Have the same slope.

• Perpendicular lines: Slopes are negative reciprocals.

Example: Find the equation of a line passing through (1, -3) and parallel to .

Pythagorean Theorem and Distance Formula

Pythagorean Theorem

In a right triangle:

Example: If and , find .

Distance and Midpoint Formulas

• Distance:

• Midpoint:

Example: Find the distance and midpoint between (2, 4) and (0, -3).

Linear Regression and Applications

Linear Regression

Linear regression finds the best-fit line for a set of data points, minimizing the sum of squared vertical distances.

• Regression Line:

• Correlation Coefficient (r): Measures the strength and direction of a linear relationship.

Example: Use the given table of credit card debt to find the regression line and predict future values.

Equations and Inequalities

Types of Equations

• Conditional Equation: Has one or more solutions, but not infinitely many.

• Contradiction: Has no solution.

• Identity: True for all values in the domain.

Example: Solve and classify the equation.

Solving Inequalities

Express solutions in interval notation.

Example: Solve .

Linear Models and Applications

Cost, Revenue, and Break-Even Analysis

• Cost Equation:

• Revenue Equation:

• Break-Even Point: Set and solve for .

Example: A T-shirt company has a fixed cost of $40,000 and a variable cost of $3 per shirt. If shirts sell for $10, find the break-even quantity.

Basic Functions and Their Graphs

Common Functions

• Linear:

• Quadratic:

• Cubic:

• Square Root:

• Cube Root:

• Absolute Value:

Continuity and Intervals

• Continuous Function: Can be drawn without lifting the pencil.

• Increasing/Decreasing/Constant: Determined by the slope of the graph over intervals.

Example: Determine intervals where a function is increasing, decreasing, or constant from its graph.

Domain and Range

Identify the set of possible x-values (domain) and y-values (range) for a function.

Symmetry and Even/Odd Functions

Symmetry

• Even Function: (symmetric about the y-axis)

• Odd Function: (symmetric about the origin)

Example: Determine if is even, odd, or neither.

Transformations of Functions

Types of Transformations

Example: The graph of can be obtained from by shifting, reflecting, and stretching/compressing.

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions over different intervals of the domain.

Example:

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

For , or .

Example: Solve .

Solving Absolute Value Inequalities

• If , then .

• If , then or .

Example: Solve and express the answer in interval notation.

Function Operations and Composition

Operations on Functions

• Sum:

• Difference:

• Product:

• Quotient: ,

Composition of Functions

The composition is defined as .

Example: If and , find and its domain.

Difference Quotient

The difference quotient is used to compute the average rate of change:

Example: For , find the difference quotient and simplify.

Additional info: Some explanations and examples were expanded for clarity and completeness, as the original material was in question format and sometimes fragmentary.