Trichotomy of Real Numbers

Types of Real Numbers

The set of real numbers can be classified into three distinct types based on their value relative to zero. This classification is fundamental in understanding the properties and behavior of numbers in algebra and precalculus.

• Negative Numbers: Any real number a such that a < 0.

• Zero: The unique real number a such that a = 0.

• Positive Numbers: Any real number a such that a > 0.

The number line visually represents these categories, with zero at the center, negative numbers to the left, and positive numbers to the right.

Linear Functions

Definition and General Form

A linear function is a function that can be written in the form:

• m is the slope of the line, representing the rate of change.

• b is the y-intercept, the value where the line crosses the y-axis.

The graph of a linear function is always a straight line.

Slope of a Line

The slope (m) of a line passing through two points and is calculated as:

• If m > 0, the function is increasing (the line rises as x increases).

• If m = 0, the function is constant (the line is horizontal).

• If m < 0, the function is decreasing (the line falls as x increases).

Domain and Range of Linear Functions

• Domain: For most linear functions, the domain is all real numbers: .

• Range: For non-horizontal lines, the range is also all real numbers: .

• For constant functions (where ), the range is a single value: .

Special Cases

• If the equation is , it represents a vertical line with undefined slope. This is not a function.

• If the equation is , it represents a horizontal line (a constant function).

Graphical Representation

Linear functions can be identified on a graph as straight lines. The direction (increasing, decreasing, or constant) depends on the sign and value of the slope.

Examples

• Example 1: For the function , the slope is 2 (increasing), and the y-intercept is 3.

• Example 2: For the function , the slope is -1 (decreasing), and the y-intercept is 1.

• Example 3: For the function , the slope is 0 (constant), and the y-intercept is 4.

Relations and Functions

Definitions

• Relation: Any set of ordered pairs .

• Function: A relation in which each input (x-value) corresponds to exactly one output (y-value).

On a graph, a function passes the vertical line test: any vertical line crosses the graph at most once.

Identifying Functions from Graphs

• Graphs that pass the vertical line test represent functions.

• Graphs that fail the vertical line test (vertical lines, circles, etc.) do not represent functions.

Domain and Range from Graphs

• Domain: The set of all possible x-values for which the function is defined.

• Range: The set of all possible y-values the function can take.

Worked Examples and Practice Problems

Finding Slope and Y-Intercept

• Given:

• To find: Slope and y-intercept.

• Solution: Rewrite in slope-intercept form (): Slope , y-intercept .

Equation from Slope and Point

• Given: Slope , point

• Equation: Use point-slope form:

Determining Linearity from Data

• Given a table of values, check if the difference in y-values is constant for equal steps in x. If so, the function is linear.

• If not, the function is non-linear.

Sample Table: Checking Linearity

Check the differences between consecutive f(x) values to determine if the function is linear.

Summary Table: Types of Linear Functions

Additional info: The above notes include expanded definitions, examples, and a summary table for clarity and completeness, as well as logical inferences about the content and context of the original materials.