Mathematics Relations and Functions

Types of Relations

• Definition: A relation is a set of ordered pairs, typically representing a relationship between two sets.
• Types:
  • One-to-One: Each element of the domain maps to a unique element in the range.
  • Onto: Every element in the range is mapped by at least one element in the domain.
  • Many-to-One: Two or more elements in the domain map to a single element in the range.
  • Many-to-Many: Elements from the domain can map to multiple elements in the range and vice versa.

Function Notation

• Definition: A function is a special type of relation where each input (from the domain) is associated with exactly one output (in the range).
• Notation: Denoted as ( f(x) ), where:
  • ( f ) represents the function name.
  • ( x ) is the input value from the domain.
  • ( f(x) ) is the corresponding output value.
• Example: If ( f(x) = 2x + 3 ), then ( f(2) = 2(2) + 3 = 7 ).

Domain and Range

• Domain:

  • The set of all possible input values (x-values) for a function.
  • Can be finite or infinite, depending on the function.
  • Examples:
    • For ( f(x) = \sqrt{x} ), the domain is ( x \geq 0 ).
    • For ( f(x) = \frac{1}{x} ), the domain is ( x \neq 0 ).

• Range:

  • The set of all possible output values (y-values) that a function can produce.
  • Determined by the function's behavior and its domain.
  • Examples:
    • For ( f(x) = x^2 ), the range is ( y \geq 0 ).
    • For ( f(x) = \sin(x) ), the range is ( -1 \leq y \leq 1 ).

Types of Relations

• A relation is defined as a set of ordered pairs representing the relationship between two sets.
• One-to-One Relation: Each element in the domain is paired with a unique element in the range, ensuring no repetitions.
• Onto Relation: Each element in the range has at least one corresponding element in the domain, allowing for complete mapping.
• Many-to-One Relation: Multiple elements from the domain link to a single element in the range, maintaining a unique output.
• Many-to-Many Relation: Elements from the domain can map to several elements in the range and vice versa, allowing for complex interactions.

Function Notation

• A function is a specific type of relation where each input from the domain is tied to exactly one output in the range.
• Function notation is expressed as ( f(x) ):
  • ( f ) indicates the function's name.
  • ( x ) is the input value chosen from the domain.
  • ( f(x) ) denotes the respective output value linked to that input.
• Example: For the function ( f(x) = 2x + 3 ), evaluating at ( x = 2 ) yields ( f(2) = 7 ).

Domain and Range

• Domain: Represents the complete set of possible input values (x-values) for a function.
  • The domain can either be finite or infinite based on the type of function applied.
  • Example domains:
    • For ( f(x) = \sqrt{x} ), the domain includes values where ( x ) is greater than or equal to zero.
    • For ( f(x) = \frac{1}{x} ), the domain excludes zero, noted as ( x \neq 0 ).
• Range: Comprises all potential output values (y-values) a function can yield.
  • Determined by the function's behavior and the constraints of its domain.
  • Example ranges:
    • For ( f(x) = x^2 ), the outputs are all non-negative, specified as ( y \geq 0 ).
    • For ( f(x) = \sin(x) ), outputs are limited to values between (-1) and (1), inclusive.