MATH 8 T3 OT1 REVIEWER

Lesson 1: Relations and Functions

Relation - A relation is a rule that describes a connection between elements of one set and another. It is composed of ordered pairs, where each first element is uniquely related to a second element. In mathematical terms, a relation can be defined as a subset of the Cartesian product of two sets.

Ordered Pairs - Ordered pairs are two values written in a specific order, typically represented as (x, y). The order is crucial, as (x, y) is not the same as (y, x); changing the order changes the meaning and the relationship.

Representing a Relation - Methods to represent a relation include:

  • Ordered Pairs: Listing pairs of related values.

  • Mapping Diagram: A visual representation where elements from Set A are connected to elements of Set B.

  • Table: A structured format listing elements in columns for Sets A and B.

  • Graph: A visual depiction with Set A on the horizontal axis (x-axis) and Set B on the vertical axis (y-axis), plotting the ordered pairs.

  • Equation: A mathematical expression representing the ordered pairs, showing the functional relationship.

Domain vs. Range

  • Domain: The domain is the set that contains all possible first coordinates (x-values) for the ordered pairs. This includes all unique x-values found in the relation.

  • Range: The range is the set containing all second coordinates (y-values) that correspond with the x-values in the domain.To find the domain, it is essential to avoid dividing by zero or taking even roots of negative numbers; these operations are undefined in the realm of real numbers. To determine the range, one can swap the x and y values, solve for y, and verify the results in the original equation by substituting back.

Lesson 2: Linear Functions

Types of Correspondence

  • One-to-One Correspondence: Each domain element pairs with exactly one unique range element, ensuring distinctiveness for every input-output relationship.

  • Many-to-One Correspondence: Several domain elements map to a single range element, often seen in functions like quadratic equations.

  • One-to-Many Correspondence: A single domain element maps to multiple range elements, indicating that the relation is not a function.

Independent vs. Dependent Variables

  • Independent Variable (x): The variable that is manipulated or selected for evaluation.

  • Dependent Variable (y): The variable whose value is determined by the independent variable, often represented by the equation y = f(x) where y depends on x.

Function Definition - A function is defined as a specific type of relation in which each x-value corresponds to a unique y-value. For example, the relation that defines the assignment between each citizen and their Personal Service Account (PSA) birth certificate serves as an excellent example of a function, demonstrating a clear and direct relationship.

Vertical Line Test - The vertical line test is a method used to determine whether a given graph represents a function: if a vertical line intersects the graph at more than one point, the relation fails the test and is therefore not a function.

Examples:

  • Every citizen to their birth certificate: This is a function.

  • Person to their birth month: This is a function as each person has one birth month.

  • Person to ice cream choices: This relationship is not a function, as one person can have multiple choices.

Lesson 3: Evaluating Functions

Definition - Evaluating a function means determining the output value (y) given a specific input (x).

Example Problem: Consider Matteo’s earnings example: his salary structure consists of a base salary of P1,200 plus P30 for each product sold: represented as y = 30x + 1200. If x = 100 (the number of products sold), then y equals P1,500.

Function Representation - Functions can be represented using notations such as f(x), g(x), h(x), and j(x) to denote various functions and their respective outputs.

Lesson 4: Basic Concepts in Geometry

Geometry Overview - Geometry is a branch of mathematics that focuses on the properties and relationships involving shapes, sizes, relative positions of figures, and the properties of space.

Undefined Terms

  • Point: Represents a precise location in space, having no dimensions (length, width, or height).

  • Line: Extends infinitely in both directions, having no thickness and composed of an infinite number of points.

  • Plane: A flat, two-dimensional surface that extends infinitely in all directions, formed by multiple points and lines.

Lesson 5: Introduction to Logic and Conditions

Proposition - A proposition is a declarative statement that is either true or false, not both.

Conditional Statement (If-Then)A conditional statement, often represented in the form p → q, includes a hypothesis (p) and a conclusion (q) thereby expressing a cause-and-effect relationship.

Truth Values - Identifying the truth values of various propositions is indispensable for evaluating the validity of conditional statements and logical arguments.

Lesson 6: Converse, Inverse, and Contrapositive Statements

Key Concepts

  • Converse: Formed by interchanging the hypothesis and conclusion of a conditional statement.

  • Inverse: Created by negating both sides of the original conditional statement.

  • Contrapositive: Formed by both interchanging and negating the hypothesis and conclusion of the original statement, often holding the same truth value as the original statement.

Lesson 7: Biconditional Statements

Definition - A biconditional statement is a compound statement that is true if and only if both the conditional and its converse are true; an example of this would be “A figure is a triangle if and only if it has three sides.”

Lesson 8: Properties of Equality

Definitions

  • Definition: The meaning of a term or phrase within a mathematical context.

  • Property: A characteristic or quality inherent to an object or entity being described.

  • Postulate: A statement accepted as true without requiring proof, serving as a foundational basis for further reasoning.

  • Theorem: A statement that has been proven based on previously established statements, axioms, and theorems.

  • Corollary: A proposition that follows readily from a previous statement or theorem.

Properties Summary: The critical properties of equality include

  • Reflexive Property: A quantity is equal to itself.

  • Symmetric Property: If one quantity equals another, then the second quantity equals the first.

  • Transitive Property: If one quantity equals a second, and the second equals a third, then the first equals the third.

  • Substitution Property: A quantity may be substituted for its equal in any expression.

  • Addition Property of Equality: If equal quantities are added to equal quantities, the sums are also equal.

  • Multiplication Property of Equality: If equal quantities are multiplied by equal quantities, the products are equal.

  • Distributive Property: A(b + c) = Ab + Ac, representing the distribution of multiplication over addition.

Lesson 9: Two-Column Proof

Importance - Finalizing a proof with a tombstone symbol (∎) indicates the proof has been completed and all asserted claims have been established through logical reasoning.

Lesson 10: Proving Segments

Key Terms

  • Congruent Segments: Segments that are equal in length, indicating equivalence in measurement.

  • Midpoint: A point that divides a segment into two equal parts, serving as the center or balance point of the segment.

  • Segment Bisector: A line, ray, or segment that intersects a segment at its midpoint, effectively dividing the segment into two equal segments.