Relations and Functions in Mathematics

Relation and Function

• A relation is a set of ordered pairs. It's a general pairing between input and output values.
• A function is a special type of relation where each input value (x-coordinate) is paired with exactly one output value (y-coordinate). No input has two or more outputs. This is the defining characteristic of a function.

Types of Relations

• Ordered pairs: A set of pairs (x, y) where x represents input and y represents output.
• Mapping diagrams: Show how inputs are mapped to outputs.
• Graphs: Visual representations of the relation on a coordinate plane.
• Sets of data: Tabular displays of input-output values.

Types of Functions

• One-to-one function: Each output (y-value) corresponds to exactly one input (x-value). A horizontal line drawn across the graph will intersect it at at most one point.
• Many-to-one function: Multiple input values (x-values) can map to the same output value (y-value). A horizontal line drawn across the graph can intersect multiple points, but each vertical line intersects at most one point.
• Onto function (surjective function): Range of function equals the codomain. Every element in the output set is mapped to by at least one input.
• Into function (not surjective function): Range of function is a proper subset of the codomain. Not every element in the output is necessarily mapped to.
• Linear function: A function whose graph forms a straight line. The most common type of function.
• Quadratic function: A function where the greatest power of x is 2; its graph is a parabola.
• Polynomial functions: Functions involving polynomials with any power.
• Rational function: Ratio of two polynomial functions.

Graphing Functions

• Cartesian coordinate system: Used to plot points and graph functions. The x-axis and y-axis intersect at the origin (0,0).
• Plotting points: Mark the ordered pairs on the graph.
• Interpreting graphs: Look for trends and relationships between x and y values.
• Finding intercepts: Where the graph crosses the x and y axes (x-intercepts and y-intercepts).

Inverse Functions

• Inverse functions "undo" one another. The inverse of a function f(x) is denoted as f^-1(x).
• To obtain the inverse, typically we swap x and y and solve for y.
• Check for invertibility: To be invertible, the original function must be one-to-one.

Function Notation

• f(x) represents the output of a function f for a given input x.
• Example: If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.
• Domain: Set of all possible input values (x-values).
• Range: Set of all possible output values (y-values).
• Codomain: The set of all possible output values the function could potentially produce, including those not actually included in the range.

This quiz covers the concepts of relations and functions, including ordered pairs, mapping diagrams, and types of functions. Understand the distinctions between one-to-one and many-to-one functions through various representations. Test your knowledge on the properties that define these mathematical concepts.