Number Sets and Classification

Types of Numbers

Numbers can be classified into several sets, each with specific properties and examples. Understanding these sets is foundational for precalculus.

• Natural Numbers: Counting numbers starting from 1. Examples: 1, 2, 3, 4, ...

• 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 , where . Examples: , , ,

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

• Real Numbers: All rational and irrational numbers.

Example: Classify the numbers , , , , , into the above sets.

Graphing Points and Intervals

Plotting Points and Quadrants

The coordinate plane is divided into four quadrants. Each point is located based on its x and y values.

• 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: Write the interval for between 1 and 5, inclusive: .

Distance and Midpoint

Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle:

Distance Formula: The distance between and is:

Midpoint Formula: The midpoint between and is:

Linear Equations and Graphs

Slope and Slope-Intercept Form

The slope of a line through and is:

• 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:

• = slope

• = y-intercept

Example: For , find the slope, y-intercept, domain, range, and zero of .

Parallel and Perpendicular Lines

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals:

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

Linear Regression

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

Example: Find the regression line and interpret the correlation coefficient.

Equations and Inequalities

Types of Equations

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

• Contradiction: No solution.

• Identity: True for all values in the domain.

Example: Solve and classify the equation.

Solving Inequalities

To solve inequalities, isolate the variable and express the solution in interval notation.

Example: Solve .

Linear Applications

Cost, Revenue, and Break-Even Analysis

• Cost equation:

• Revenue equation:

• Break-even point: Where

Example: If fixed cost is and variable cost is $3, find the break-even quantity.

Functions and Their Properties

Basic Functions

• Linear function:

• Square function:

• Cube function:

• Square root function:

• Cube root function:

• Absolute value function:

Continuity, Domain, and Range

A function is continuous if its graph can be drawn without lifting the pencil. The domain is the set of all possible input values, and the range is the set of all possible output values.

Example: Determine the domain and range of .

Increasing, Decreasing, and Constant Intervals

• Increasing: for

• Decreasing: for

• Constant: for

Symmetry, Even and Odd Functions

• 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 is obtained from by shifting left 2 units, reflecting across the x-axis, vertically shrinking by a factor of , and shifting down 3 units.

Piecewise Functions

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

Example:

Absolute Value Equations and Inequalities

Absolute Value Equations

For , or .

Example: Solve .

Absolute Value Inequalities

• If , then

• If , then or

Example: Solve and express the answer in interval notation.

Operations with Functions

Function Operations

• Sum:

• Difference:

• Product:

• Quotient: ,

• Composition:

Example: If and , find , , and their domains.

Difference Quotient

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

Example: For , find and simplify the difference quotient.

Additional Applications

• Modeling cost, revenue, and profit with linear functions

• Solving real-world problems using systems of equations

• Analyzing graphs for domain, range, and intervals of increase/decrease

Example: Suppose you have to invest in stocks and bonds. Stocks pay annually, bonds pay . If the annual interest income is , how much was invested in each?