inverse function

Declared original function

Record the original rule and its domain and codomain, distinguishing the codomain from the actual image.

1. What is the original function f:A→B, including its rule and explicitly declared sets A and B? definition
2. Does f attain every element of B, or is the proposed inversion actually limited to f(A)? boundary

Candidate inverse identity

Record the reverse mapping and establish that it denotes a function inverse rather than a reciprocal or preimage operator.

1. What mapping g is proposed as the inverse, and do its domain and codomain reverse those of the specified original mapping? definition
2. What definition or contextual evidence establishes that f⁻¹ denotes an inverse function here? provenance

Unique preimage coverage

Record evidence for injectivity and surjectivity on the declared sets, including counterexamples and unresolved conditions.

1. What establishes that distinct inputs in A cannot produce the same output under f? definition
2. What establishes that every intended inverse input has a preimage, and which inputs remain excluded or unresolved? boundary
3. Which proof, theorem with checked hypotheses or exhaustive finite check supports these conclusions? provenance

Inverse identities

Record the status and extent of evidence for g∘f=id_A and f∘g=id_B.

1. Does g(f(x))=x hold for every x in A, including any included boundary points? boundary
2. Does f(g(y))=y hold for every y in B, and is this established exactly or only tested on samples? provenance

Single-valued restriction

Identify the restricted original function and the reason for selecting its inverse over other possible branches.

1. If f is not injective on its original domain, which subset A' is selected and what is its image f(A')? boundary
2. What task requirement or stated convention selects this restriction rather than another valid inverse branch? provenance

Branch validity region

Record valid neighbourhoods, included endpoints, excluded values and any branch cuts relevant to the mapping.

1. On which exact sets or neighbourhoods is this inverse valid, and which endpoints or cut boundaries are included? boundary
2. What must an agent check before extending the inverse's domain or switching to another branch? action

Representation validity

Record how the representation was derived and why it denotes the selected inverse throughout its claimed domain.

1. Is the inverse represented explicitly, implicitly, algorithmically or by a finite lookup? definition
2. What derivation or verification connects this representation to the original function and selected branch? provenance

Evaluation result and error

Record input admissibility, output accuracy and computational failure conditions without equating a small residual with a small inverse error.

1. What domain, branch, initialisation or convergence checks must pass before the evaluation result can be accepted? action
2. What bound supports the error in the returned inverse value, and what assumptions connect any residual |f(x̂)−y| to that error? measurement

Supported inverse behaviour

Record which regularity properties hold and where recovering an input becomes sensitive to perturbations.

1. Which continuity or differentiability properties of the inverse are established, and on what sets and under which hypotheses? boundary
2. Where is inverse evaluation sensitive to perturbations, and what declared derivative, norm or condition measure supports that assessment? measurement

Operation preconditions

Record when applying the inverse solves the intended equation or supports a transformation without losing branch or domain conditions.

1. Before replacing f(x)=y with x=g(y), how will the agent confirm that y lies in the inverse domain and x is constrained to the selected original domain? action
2. Which additional hypotheses must be established for the intended use, such as differentiation or a change of variables? action

Kinds and varieties

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• two-sided inverse of a bijection
• left inverse of an injection
• right inverse of a surjection
• local inverse (inverse-function theorem)
• principal branch of a multivalued inverse
• inverse trigonometric functions
• inverse hyperbolic functions
• inverse of an analytic function (Lagrange inversion)

1. Which of these kinds and varieties hold for the sense of inverse function this model covers, and on what evidence? provenance

Identifiers and schemes

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• Wikidata - Q191884 - Item for inverse function / inverse mapping; distinct from Q3075242 (reciprocal function x ↦ 1/x).
• ISO 80000-2:2019 - 2-12.10 - Named 'inverse function of f'; symbol f^{-1}.
• GND - 4186767-1 - Deutsche Nationalbibliothek subject heading linked from Wikidata.
• Freebase / Google Knowledge Graph - /m/03vjb - Freebase mid recorded on Wikidata P646.
• Wolfram Language - InverseFunction[f] - Computer-algebra constructor for a symbolic inverse.
• Encyclopædia Britannica topic - topic/inverse-function - Britannica identifier recorded on Wikidata P1417.

1. Which of these identifiers and schemes hold for the sense of inverse function this model covers, and on what evidence? provenance

Standards and regulation

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• ISO 80000-2:2019 Quantities and units - Part 2: Mathematics, item 2-12.10 (International Organization for Standardization): defines f^{-1} for injective f, sets dom(f^{-1})=ran(f), and forbids reading f^{-1} as the pointwise reciprocal.
• ISO 80000-2:2019 (same part): since the 2019 revision, inverse trigonometric functions are to be written with the arc- prefix (arcsin, arccos, …), not as sin^{-1}.

1. Which of these standards and regulation hold for the sense of inverse function this model covers, and on what evidence? provenance

Real-world use

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• Undoing a named elementary map: logarithm inverts exponentiation, square root inverts squaring on [0,∞), arcsin recovers an angle from a sine ratio in navigation, surveying and computer graphics.
• Change of variables and substitution in calculus, and recovering a coordinate chart locally when the Jacobian is non-singular (inverse-function theorem).
• Computer-algebra and numeric libraries expose InverseFunction, asin/acos/atan (and atan2) as the everyday interface; branch cuts are part of the contract.
• Cryptography and one-way hashing treat invertibility as a security property: a function is used precisely because a feasible inverse is not known.
• Control, signal processing and inverse kinematics invert a forward map (plant, filter, or robot Jacobian) to recover an input or joint angle.

1. Which of these real-world use hold for the sense of inverse function this model covers, and on what evidence? provenance

Typical measurements

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• existence of a two-sided inverse - yes iff f is bijective; otherwise no - dimensionless (boolean)
• Jacobian determinant det Df(x) as local-invertibility witness - nonzero real at points of local invertibility; zero at critical points - dimensionless (or product of the coordinate units of f)
• derivative of the inverse, (f^{-1})'(y)=1/f'(x) for y=f(x) - unbounded as f'(x)→0; order-1 when |f'| is order-1 - reciprocal of the unit of f'
• principal range of real arcsine - [-π/2, π/2] - radian
• principal range of real arccosine - [0, π] - radian
• real domain of arcsine / arccosine - [-1, 1] - dimensionless

1. Which of these typical measurements hold for the sense of inverse function this model covers, and on what evidence? provenance

Failure modes and hazards

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• Writing f^{-1} when f is not injective (classic case: x ↦ x² on ℝ), so the converse relation is not a function.
• Reading f^{-1} as the pointwise reciprocal 1/f, which ISO 80000-2 and calculus texts both flag; sin^{-1}x is the usual collision.
• Taking a right inverse for a two-sided inverse: √(x²) equals |x|, not x.
• Crossing a branch cut or leaving the principal range of a multivalued inverse (complex log, arcsin, square root), so f(f^{-1}(z))=z or f^{-1}(f(z))=z fails.
• Inverse-function theorem does not apply where the Jacobian vanishes, so no C¹ local inverse exists through a critical point.
• Numerical inversion becomes ill-conditioned as f'→0; library asin/acos raise domain errors outside [-1,1].

1. Which of these failure modes and hazards hold for the sense of inverse function this model covers, and on what evidence? provenance

Regional variation

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• French: fonction réciproque is the inverse function; fonction inverse is the reciprocal x ↦ 1/x (Wikidata Q3075242). Spanish usage sometimes follows that pair (función recíproca on the inverse-function item).
• German Umkehrfunktion and Norwegian invers funksjon / omvendt funksjon are the ordinary names; English anti-function is attested but rare.
• Notation split: ISO 80000-2:2019 requires arcsin etc.; English school and programming practice still uses sin^{-1}, asin, acos, atan.
• Set-theory texts demand a bijection onto the stated codomain for a two-sided inverse; ISO 80000-2:2019 and many calculus texts define f^{-1} as soon as f is injective, taking the domain of the inverse to be the image.

1. Which of these regional variation hold for the sense of inverse function this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Reported by the breadth pass; each item needs checking against its source before it becomes normative.

• reciprocal (multiplicative inverse) function x ↦ 1/x - The reciprocal is pointwise multiplication-inverse; the inverse function is composition-inverse. Test: check whether g(f(x))=x or whether f(x)·g(x)=1.
• preimage / inverse image f^{-1}({y}) - The preimage is a set and may have any cardinality; an inverse function exists only when every relevant preimage is a singleton. Test: count solutions of f(x)=y.
• inverse relation (converse of a binary relation) - Every function has a converse relation; that converse is an inverse function only if it is itself a function. Test: is the converse single-valued and total on the image?
• inverse of a matrix / linear map - Matrix inverse is the inverse function in the special case where f is linear between finite-dimensional spaces. Test: is f linear? If not, no matrix represents it.
• inverse problem (recovering causes from observed effects) - An inverse problem is a reconstruction task, often ill-posed; it is not the same object as the inverse of a given function. Test: is there a specified map f whose compositional inverse is claimed, or a data-to-cause inference?
• antiderivative - Differentiation and indefinite integration are inverse operations up to a constant, not inverse functions of a single map. Test: is uniqueness required, and is the object a function or a family of primitives?

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of inverse function this model covers, and on what evidence? provenance