Types of Functions

A function f : A → B is called one-one (or injective) if distinct elements of A always map to distinct elements of B. Formally: if f(x₁) = f(x₂) then x₁ = x₂, equivalently if x₁ ≠ x₂ then f(x₁) ≠ f(x₂). Example: f(x) = 3x + 1 on ℝ is one-one because if 3x₁ + 1 = 3x₂ + 1 then x₁ = x₂. On a graph, a function is one-one if and only if every horizontal line crosses the graph at most once (Horizontal Line Test).

A function f : A → B is many-one if at least two distinct elements of A share the same image in B. That is, there exist x₁ ≠ x₂ in A with f(x₁) = f(x₂). Example: f(x) = x² on ℝ is many-one because f(2) = f(−2) = 4. Trigonometric functions like sin x and cos x are also many-one over ℝ. A many-one function fails the Horizontal Line Test — some horizontal line cuts the graph more than once. Many-one functions do NOT have inverses unless their domain is restricted.

A function f : A → B is onto (or surjective) if every element b ∈ B is the image of at least one element a ∈ A. Formally: for every b ∈ B there exists a ∈ A such that f(a) = b. The range of f equals its entire codomain B. Example: f(x) = 2x − 1 from ℝ to ℝ is onto because for any y ∈ ℝ we can solve: a = (y + 1)/2 ∈ ℝ, and f(a) = y. However, f(x) = x² from ℝ to ℝ is NOT onto because negative numbers are never outputs.

A function f : A → B is called into if it is NOT onto, meaning its range is a proper subset of B. At least one element of B is never mapped to by any element of A. Example: f(x) = x² from ℝ to ℝ is into because the range is [0, ∞) ⊊ ℝ. Note that every function is either onto or into — these are complementary categories. The distinction matters when asking whether an equation f(x) = c always has a solution.

A function f : A → B is bijective if it is both injective and surjective simultaneously. Every element of B is the image of exactly one element of A — perfect pairing with no element left out and no element matched twice. Example: f(x) = 5x − 3 from ℝ to ℝ is bijective. Bijective functions always have a well-defined inverse function f⁻¹ : B → A. If A and B are finite sets, bijectivity implies |A| = |B|. The identity function f(x) = x is always bijective on any set.

A constant function maps every element of A to the same single element c ∈ B: f(x) = c for all x ∈ A. If |A| > 1, a constant function is always many-one (all inputs share one output). It is onto only when |B| = 1. Its graph is a horizontal line. Although simple, constant functions appear naturally as limiting cases and in piecewise definitions.

• f — Function (none): The rule assigning each element of A to an element of B
• A — Domain (none): The set of all valid inputs for the function
• B — Codomain (none): The set containing all possible outputs (the target set)
• x_1, x_2 — Input elements (none): Arbitrary elements chosen from the domain A
• \implies — Implies (none): Logical implication — if the left side is true, so is the right side

Problem. Let f : ℝ → ℝ be defined by f(x) = 2x² − 3. Determine whether f is (a) one-one, (b) onto.

Answer. f is many-one and into (neither injective nor surjective over ℝ).

• Inverse Functions — Only bijective functions (one-one and onto) have a well-defined inverse function. Understanding function types is a prerequisite for studying inverses.
• Domain and Range — Onto functions are defined by the range equalling the entire codomain, making domain and range concepts directly tied to classifying surjectivity.
• Definition of a Function — Understanding what makes a relation a function is the logical first step before classifying functions into types such as injective or surjective.
• Composition of Functions — Injectivity and surjectivity behave predictably under composition — e.g. the composition of two injective functions is injective — linking these types to composite functions.
• Pigeonhole Principle — The pigeonhole principle is deeply connected to injections: if |A| > |B|, no injective function from A to B can exist, making function types central to combinatorial arguments.