Grade 9 MATH Term 1 Week 3: Relations & Functions, Domain & Range | MATATAG First Term/Q1 Tagalog

The lesson begins by introducing the concept of relations as pairings, connections, or relationships between two quantities or objects, using everyday examples like a student's section, advisor, or favorite subjects. It then defines a function as a special type of relation where each input has exactly one output, likening it to a machine where one input yields one output. A key distinction is made: all functions are relations, but not all relations are functions.

The video explains five ways to represent relations: ordered pairs (input/output), table of values (x and y columns), mapping diagrams (visual representation of pairings), graphs (using x-axis for inputs and y-axis for outputs), and rules/equations (algebraic expressions like y = 2x). It then demonstrates how to identify functions from these representations. For ordered pairs and tables, a relation is a function if no input (x-value) is paired with more than one output (y-value). Mapping diagrams distinguish between one-to-one (function), one-to-many (not a function), and many-to-one (function) relationships. The vertical line test is introduced for graphs, stating that if a vertical line intersects the graph at only one point, it is a function.

Domain is defined as the set of all possible input (x) values, and range as the set of all possible output (y) values. The video shows how to find the domain and range from ordered pairs, tables, and mapping diagrams by simply listing the unique x-values (domain) and y-values (range). It emphasizes not repeating values when listing elements of a set.

For functions represented by equations, specific rules apply. For linear functions (e.g., y = x + 2), the domain and range are typically all real numbers. For rational functions (e.g., y = 2/(x-3)), the domain excludes x-values that make the denominator zero, and the range excludes y-values that are not possible outputs. For quadratic functions (e.g., y = x^2 + 2), the domain is usually all real numbers, while the range is restricted based on the vertex of the parabola. For absolute value functions (e.g., y = |x|), the domain is all real numbers, but the range is non-negative numbers (y >= 0).

The video illustrates how to determine domain and range directly from graphs. For continuous lines with arrowheads, both domain and range are often all real numbers. For parabolas, the domain is all real numbers, while the range is limited by the vertex. For graphs with endpoints, the domain and range are defined by the minimum and maximum x and y values, respectively, as indicated by the solid or open circles at the endpoints.

This section explains how to express a relationship in functional notation. Independent variables (inputs, typically x) are those whose values do not depend on others, while dependent variables (outputs, typically y) depend on the independent variables. The functional notation f(x) = y (read as 'f of x equals y') is introduced. Examples demonstrate how to identify independent and dependent variables, and then formulate an equation or rule that represents the relationship as a function, and how to use it to solve problems.

The video concludes by reiterating the connection between real-life relationships and mathematical functions, emphasizing that functions are special relations where each input leads to exactly one output. It draws a parallel to personal choices and connections, suggesting that positive connections and good habits lead to positive outcomes in life, just as inputs are connected to outputs in relations.