2.2 Functions on Sets

Functions on sets are the backbone of mathematical relationships. They define how elements from one set map to another, establishing crucial connections in various mathematical and real-world scenarios.

This topic explores different types of functions, their properties, and operations. Understanding these concepts is essential for grasping more complex mathematical ideas and solving problems in fields like computer science and data analysis.

Function Basics

Understanding Domain and Codomain

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• Domain represents the set of all possible input values for a function
• Consists of all valid x-values that can be used in the function
• Codomain encompasses all possible output values of a function
• Includes every y-value that could potentially result from the function
• Domain and codomain form the foundation for defining a function's behavior
• Can be finite or infinite sets depending on the function's nature

Range and Its Relationship to Codomain

• Range constitutes the set of actual output values produced by a function
• Always a subset of the codomain, may be equal to or smaller than the codomain
• Determined by applying the function to every element in the domain
• Helps identify which codomain elements are actually used by the function
• Range analysis reveals important characteristics of the function's behavior

Identity Function and Its Properties

• Maps each element of a set to itself
• Defined as f(x) = x for all x in the domain
• Serves as a neutral element in function composition
• Preserves the structure and properties of the input set
• Often used as a reference point in mathematical proofs and constructions
• Identity function for real numbers: f(x) = x, where x is any real number

Function Types

Injective Functions (One-to-One)

• Each element of the codomain is mapped to by at most one element of the domain
• No two distinct elements in the domain map to the same element in the codomain
• Can be visualized as each input having a unique output
• Horizontal line test used to determine if a function is injective
• Ensures reversibility of the function within its range
• Example: f(x) = 2x for real numbers is injective, as each output corresponds to a unique input

Surjective Functions (Onto)

• Every element in the codomain is mapped to by at least one element in the domain
• Range of the function equals its codomain
• Ensures that the function covers the entire codomain
• Can have multiple elements in the domain mapping to the same codomain element
• Vertical line test used to determine if a function is surjective
• Example: f(x) = x² for real numbers to non-negative real numbers is surjective

Bijective Functions and Their Significance

• Combines properties of both injective and surjective functions
• Each element in the codomain is mapped to by exactly one element in the domain
• Establishes a one-to-one correspondence between domain and codomain
• Guarantees the existence of an inverse function
• Crucial in establishing isomorphisms between mathematical structures
• Example: f(x) = 3x + 1 for real numbers is bijective, as it's both injective and surjective

Function Operations

Inverse Functions and Their Properties

• Reverses the effect of the original function
• Exists only for bijective functions
• Denoted as f⁻¹(x) for a function f(x)
• Switches the roles of domain and codomain
• Composition of a function with its inverse yields the identity function
• Finding inverse involves solving for x in terms of y and then swapping x and y
• Example: If f(x) = 2x + 3, then f⁻¹(x) = (x - 3) / 2

Function Composition and Its Applications

• Combines two or more functions to create a new function
• Denoted as (f ∘ g)(x) or f(g(x)), where g is applied first, then f
• Domain of g must be compatible with the codomain of f
• Not always commutative: (f ∘ g) ≠ (g ∘ f) in general
• Associative property applies: (f ∘ g) ∘ h = f ∘ (g ∘ h)
• Used extensively in mathematical modeling and computer programming
• Example: If f(x) = x² and g(x) = x + 1, then (f ∘ g)(x) = (x + 1)²