Functions glossary

Clear, one-line definitions of the Functions terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

bijection

A function that is both an injection and a surjection; it pairs domain and codomain one-to-one.

bijective

A function that is both injective (one-to-one) and surjective (onto); equivalently, has an inverse.

codomain

The set in which a function's outputs are required to lie.

composition

The function (g ∘ f)(x) = g(f(x)) formed by applying f then g.

See: Composition and Inverse Mappings

composition of functions

(g ∘ f)(x) = g(f(x)); apply f first, then g.

domain

The set of all valid inputs to a function.

See: Frequency-domain (Helmholtz) formulation, Time vs depth: the two domains of seismic

domain (of a function)

The set of inputs A on which a function f: A → B is defined.

exponential function

A function of the form f(x) = a^x with a > 0, a ≠ 1; grows or decays at a rate proportional to its current value.

See: Exponential Functions

function

A rule that assigns to each element of a domain exactly one element of a codomain.

See: Functions, What Is a Function?

identity function

id_A: A → A defined by id_A(x) = x for every x ∈ A.

image

The image of an element a under a mapping f is f(a); the image of a function is the set {f(a) : a ∈ domain}.

image of a function

The set of all outputs the function actually produces; a subset of the codomain.

injection

A function for which distinct inputs always produce distinct outputs; also called one-to-one.

See: 4D time-lapse monitoring at a Sleipner CO2-injection target

injective

A function f is injective (one-to-one) when f(a) = f(b) implies a = b; distinct inputs give distinct outputs.

inverse

For a bijective f: A → B, the inverse f⁻¹: B → A satisfies f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.

See: Inverse Isometries, Inverses of Functions

inverse function

For a bijection f, the function f⁻¹ that undoes f: f⁻¹(f(x)) = x.

See: The Inverse Function Theorem

inverse image (preimage)

For f: A → B and T ⊆ B: f⁻¹(T) = {x ∈ A : f(x) ∈ T}; inputs that map into T.

logarithm

The exponent y to which a base b must be raised to give x; written y = log_b(x).

mapping

A synonym for function, emphasizing the assignment of inputs to outputs.

See: Composition and Inverse Mappings, Mappings as Generalized Functions

range (of a function)

For f: A → B, the range is {f(x) : x ∈ A} ⊆ B, the set of actual outputs.

surjection

A function whose image is the entire codomain; every codomain element is hit.

surjective

A function f: A → B is surjective (onto) when its range equals its codomain; every b ∈ B has a preimage.