General Mathematics Lecture #1: Relations & Functions

Key Concepts in Relations & Functions

• A relation is defined as a set of ordered pairs (x, y).
• Example of a relation: A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}.
• Ordered pairs are also known as coordinates, where:
  • Abscissa refers to the x-value.
  • Ordinate refers to the y-value.

Domain and Range

• Domain: The set of all x-values in a relation.
  • Example: Domain of set A = {1, 2, 3, 4}.
• Range: The set of all y-values in a relation.
  • Example: Range of set A = {3, 5, 9, 11, −15}.

Determining Domain and Range

• Various relations provided for practice:
  • A = {(−3, −4), (−2, 5), (−1, 5), (0, 6)}.
  • B = {(−2, 5), (0, 5), (2, 5), (4, 5)}.
  • C = {(0, 3), (0, 6), (−2, −4), (−4, −5)}.
  • D = {(1, 0), (0, 1), (−1, 0), (0, −1)}.
  • E = {(-4, 2), (-1, 3), (-4, 5), (-1, 0)}.

Understanding Functions

• A function is a special type of relation where no two distinct ordered pairs have the same first element (x-value).
• All functions are relations, but not all relations qualify as functions.

Ways to Represent a Function

• Numerically:
  • No shared x-coordinates among ordered pairs.
  • Representation through ordered pairs or a table of values.
• Graphically:
  • Use the vertical line test; a function's graph will intersect any vertical line at exactly one point.
• Mapping:
  • Types include one-to-one, one-to-many, many-to-one, and many-to-many correspondences.
• Algebraically:
  • The exponent of the y variable must not exceed one.
  • Example: Equation representation, such as 2x − 3y = 6.
• Verbally:
  • Formulated rules of correspondence expressed in words (like the relation between height and shoe size).

Function Verification

• A relation is a function if:
  • No two distinct pairs share the same x-coordinate.
  • A vertical line intersects the graph at one point.
  • In mappings, only one-to-one and many-to-one are considered functions.
  • Algebraic representation satisfies the degree condition of y.
  • Rules of correspondence indicate singularity in terms.

Classification of Relations and Functions

• Analyze given sets and equations to deem whether they are relations or functions.
• Examples provided for classification include:
  • A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}.
  • Various mathematical relations and their contextual interpretations such as plate numbers or student references.
• Apply criteria to equations like y = √(x² − 1) or polynomials to assess if they define functions.