Functions, Cartesian Plane and Relations

Study notes on Functions, the Cartesian Plane, Relations, Graphing Functions, Coordinate Systems, and Relation Properties:

• Functions, the Cartesian Plane, Relations, Graphing Functions, Coordinate Systems, and Relation Properties form the foundation for understanding mathematical relationships and their visual representations.

Functions

• A function is a relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output.
• The input set is called the domain, and the set of possible output values is called the range of the function.
• Functions can be represented through equations, graphs, or tables.
• The vertical line test determines if a graph represents a function, with a vertical line intersecting the graph at most once.
• Notation: f(x) represents the output of the function f for the input x.

Cartesian Plane

• The Cartesian plane is a two-dimensional coordinate system defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
• Points are identified by ordered pairs (x, y), where x is the horizontal distance from the origin (0, 0) along the x-axis, and y is the vertical distance from the origin along the y-axis.
• The plane is divided into four quadrants, numbered I to IV, based on the signs of the x and y coordinates.
• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y > 0

Relations

• A relation is a set of ordered pairs (x, y).
• The set of all first elements (x-values) is called the domain of the relation, and the set of all second elements (y-values) is called the range of the relation.
• Relations can be represented in various ways, including sets of ordered pairs, tables, graphs, and equations.
• Not all relations are functions; a relation is a function only if each x-value is associated with exactly one y-value.
• Relations can be used to describe any kind of correspondence between two sets.

Graphing Functions

• Graphing a function involves plotting ordered pairs (x, f(x)) on the Cartesian plane.
• The graph provides a visual representation of the connection between the input and output values of the function.
• Key features of a graph include intercepts (where the graph crosses the x or y axes), slope, maximum and minimum points, and asymptotes.
• Different types of functions (linear, quadratic, exponential, etc.) have characteristic graph shapes.
• Transformations of functions (translations, reflections, stretches, and compressions) can be visually represented by changes in the graph.

Coordinate Systems

• Besides the Cartesian coordinate system, other coordinate systems exist, such as polar coordinates.
• Polar coordinates represent a point in the plane by its distance r from the origin and the angle θ it makes with the positive x-axis.
• Conversion between Cartesian and polar coordinates:
  • x = r * cos(θ)
  • y = r * sin(θ)
  • r^2 = x^2 + y^2
  • θ = arctan(y/x)
• Polar coordinates can be useful for representing functions with circular symmetry.
• Three-dimensional coordinate systems extend the Cartesian plane by adding a z-axis, allowing for the representation of points in space with ordered triples (x, y, z).

Relation Properties

• Relations can have properties such as reflexivity, symmetry, antisymmetry, and transitivity.
• A relation R on a set A is reflexive if (a, a) ∈ R for all a ∈ A.
• A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b ∈ A.
• A relation R on a set A is antisymmetric if (a, b) ∈ R and (b, a) ∈ R implies a = b for all a, b ∈ A.
• A relation R on a set A is transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R for all a, b, c ∈ A.
• These properties are important in various areas of mathematics, including set theory, graph theory, and database design.