Mathematics Relations and Functions

Study Notes

Relations

Definition: A relation is a set of ordered pairs (x, y) where x is from set X (domain) and y is from set Y (range).
Types of Relations:
- One-to-One: Each element of the domain is paired with a unique element of the range.
- Many-to-One: Multiple elements of the domain are paired with the same element in the range.
- One-to-Many: A single element of the domain is paired with multiple elements in the range (not a function).
- Many-to-Many: Multiple elements of the domain are paired with multiple elements in the range (not a function).
Properties:
- Reflexive: Every element is related to itself (aRa).
- Symmetric: If aRb, then bRa.
- Transitive: If aRb and bRc, then aRc.
Graphical Representation: Relations can be visualized using graphs, where points represent ordered pairs.

Functions

Definition: A function is a special type of relation where each element in the domain is associated with exactly one element in the range.
Notation: Typically represented as f(x), where f is the function and x is the input from the domain.
Domain and Range:
- Domain: Set of all possible input values.
- Range: Set of all possible output values.
Types of Functions:
- Linear Functions: Represented by f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic Functions: Represented by f(x) = ax^2 + bx + c, forming a parabola.
- Exponential Functions: Represented by f(x) = ab^x, where a is a constant and b is the base.
- Polynomial Functions: Functions with terms of non-negative integer powers of x.
- Rational Functions: Functions defined by the ratio of two polynomials.
Vertical Line Test: A graph represents a function if no vertical line intersects the graph in more than one point.
Inverse Functions: If f(x) pairs each x with a unique y, the inverse function f⁻¹(y) pairs each y back with its corresponding x.

Relations

A relation is defined as a set of ordered pairs (x, y) from set X (domain) and set Y (range).
Types of Relations:
- One-to-One: Each domain element links to a unique range element, ensuring no duplicates.
- Many-to-One: Several domain elements correspond to a single range element, creating possible overlaps.
- One-to-Many: A single domain element relates to multiple range elements, this is not classified as a function.
- Many-to-Many: Multiple domain elements are related to multiple range elements, also not a function.
Properties:
- Reflexive: Indicates that every element relates to itself, expressed as aRa.
- Symmetric: If a is related to b (aRb), then b must also relate back to a (bRa).
- Transitive: If a relates to b (aRb) and b relates to c (bRc), then a must relate to c (aRc).
Graphical representations illustrate relations using points to depict ordered pairs.

Functions

A function is a specialized relation where each domain element corresponds to exactly one range element.
Notation: Functions are typically written as f(x), indicating f is the function and x is the input from the domain.
Domain and Range:
- Domain: Represents all possible inputs for the function.
- Range: Comprises all possible outputs produced from the inputs.
Types of Functions:
- Linear Functions: Defined by f(x) = mx + b, with m as the slope and b as the y-intercept, graphing as a straight line.
- Quadratic Functions: Expressed as f(x) = ax^2 + bx + c, characterized by a parabolic shape.
- Exponential Functions: Written as f(x) = ab^x, where a is a constant and b serves as the base of the exponent.
- Polynomial Functions: Involves terms with non-negative integer powers of x.
- Rational Functions: Consist of the ratio of two polynomials where the denominator is not zero.
The Vertical Line Test determines if a graph represents a function by ensuring no vertical line intersects the graph in more than one location.
Inverse Functions: For a function f(x) that pairs x with a unique y, the inverse function f⁻¹(y) reverses the roles, linking each y back to its corresponding x.

Description

Explore the concepts of relations and functions in mathematics through this quiz. Understand types of relations such as one-to-one and many-to-many, along with their properties like reflexive and transitive. Test your knowledge on graphical representations and the distinctions between relations and functions.