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Question 32755: Inverse Value at a Permitted Target

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0064-32755 · q-20261004-advance-20261005-r04-inverse-function-0030

Original question and givens

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 5∈B, find g(5).

  • G001 — The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 5∈B, find g(5).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-inverse-function-evaluation, inverse_evaluate, no-required-domain, []]

After

derived = [[unique_preimage_integer_STRING]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependencies

Rule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001

S002 · implies

Before

source roles = [ORIGINAL f domain, [-6, -5, -4, -3, -2, -1]]

After

derived = [[-6, -5, -4, -3, -2, -1]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S001

S003 · implies

Before

source roles = [ORIGINAL f codomain, [1, 3, 5, 7, 9, 11]]

After

derived = [[1, 3, 5, 7, 9, 11]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S002

S004 · implies

Before

source roles = [ORIGINAL f, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

Conditions: none

FIRST type EVERY raw pair and enforce raw length≤200,THEN normalize identical rows as one ordered pair.Do not merge different outputs of one input.Empty domains do not waive RAW checks.

Step identifiers and dependencies

Rule: FiniteMap_RAW200_bound_BEFORE_identical_row_ASSET_normalization · Uses: G001, S003

S005 · implies

Before

source roles = [ORIGINAL f, 0, -6, 11, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S004

S006 · implies

Before

source roles = [ORIGINAL f, 1, -5, 1, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S005

S007 · implies

Before

source roles = [ORIGINAL f, 2, -4, 3, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S006

S008 · implies

Before

source roles = [ORIGINAL f, 3, -3, 5, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S007

S009 · implies

Before

source roles = [ORIGINAL f, 4, -2, 7, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S008

S010 · implies

Before

source roles = [ORIGINAL f, 5, -1, 9, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S009

S011 · implies

Before

source roles = [ORIGINAL f, -6, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S010

S012 · implies

Before

source roles = [ORIGINAL f, -5, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S011

S013 · implies

Before

source roles = [ORIGINAL f, -4, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S012

S014 · implies

Before

source roles = [ORIGINAL f, -3, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S013

S015 · implies

Before

source roles = [ORIGINAL f, -2, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S014

S016 · implies

Before

source roles = [ORIGINAL f, -1, [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S015

S017 · implies

Before

source roles = [ORIGINAL f, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes, Yes], [1, 1, 1, 1, 1, 1], Yes]

Conditions: none

Permit lookups ONLY after BOTH all-row typing and all-input totality.Empty-set vacuity cannot conceal an invalid row.

Step identifiers and dependencies

Rule: FiniteMap_TOTAL_qualification_BEFORE_lookups_even_empty_domain · Uses: G001, S016

S018 · implies

Before

source roles = [ORIGINAL f, 1, [-6, -5, -4, -3, -2, -1], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[-5], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S017

S019 · implies

Before

source roles = [ORIGINAL f, 3, [-6, -5, -4, -3, -2, -1], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[-4], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S018

S020 · implies

Before

source roles = [ORIGINAL f, 5, [-6, -5, -4, -3, -2, -1], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[-3], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S019

S021 · implies

Before

source roles = [ORIGINAL f, 7, [-6, -5, -4, -3, -2, -1], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[-2], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S020

S022 · implies

Before

source roles = [ORIGINAL f, 9, [-6, -5, -4, -3, -2, -1], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[-1], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S021

S023 · implies

Before

source roles = [ORIGINAL f, 11, [-6, -5, -4, -3, -2, -1], [[-6, 11], [-5, 1], [-4, 3], [-3, 5], [-2, 7], [-1, 9]]]

After

derived = [[-6], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S022

S024 · implies

Before

source roles = [ORIGINAL f, [-6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11], [[-5], [-4], [-3], [-2], [-1], [-6]]]

After

derived = [Yes, Yes, Yes]

Conditions: none

Injective means every fiber≤1,surjective every fiber≥1,and BOTH mean bijective.Unattained B does not defeat injectivity.Empty→empty is bijective;empty→nonempty is only injective.

Step identifiers and dependencies

Rule: FiniteMap_injective_surjective_BOTH_needed_for_inverse · Uses: G001, S023

S025 · implies

Before

source roles = [-6, 11, 5, [1, 3, 5, 7, 9, 11]]

After

derived = [No]

Conditions: none

FIRST require t∈B,THEN inspect EVERY A input for f(x)=t.This is inverse evaluation,not f(t)or1/f(t).

Step identifiers and dependencies

Rule: Inverse_EVALUATE_EACH_original_input_matches_typed_target · Uses: G001, S024

S026 · implies

Before

source roles = [-5, 1, 5, [1, 3, 5, 7, 9, 11]]

After

derived = [No]

Conditions: none

FIRST require t∈B,THEN inspect EVERY A input for f(x)=t.This is inverse evaluation,not f(t)or1/f(t).

Step identifiers and dependencies

Rule: Inverse_EVALUATE_EACH_original_input_matches_typed_target · Uses: G001, S025

S027 · implies

Before

source roles = [-4, 3, 5, [1, 3, 5, 7, 9, 11]]

After

derived = [No]

Conditions: none

FIRST require t∈B,THEN inspect EVERY A input for f(x)=t.This is inverse evaluation,not f(t)or1/f(t).

Step identifiers and dependencies

Rule: Inverse_EVALUATE_EACH_original_input_matches_typed_target · Uses: G001, S026

S028 · implies

Before

source roles = [-3, 5, 5, [1, 3, 5, 7, 9, 11]]

After

derived = [Yes]

Conditions: none

FIRST require t∈B,THEN inspect EVERY A input for f(x)=t.This is inverse evaluation,not f(t)or1/f(t).

Step identifiers and dependencies

Rule: Inverse_EVALUATE_EACH_original_input_matches_typed_target · Uses: G001, S027

S029 · implies

Before

source roles = [-2, 7, 5, [1, 3, 5, 7, 9, 11]]

After

derived = [No]

Conditions: none

FIRST require t∈B,THEN inspect EVERY A input for f(x)=t.This is inverse evaluation,not f(t)or1/f(t).

Step identifiers and dependencies

Rule: Inverse_EVALUATE_EACH_original_input_matches_typed_target · Uses: G001, S028

S030 · implies

Before

source roles = [-1, 9, 5, [1, 3, 5, 7, 9, 11]]

After

derived = [No]

Conditions: none

FIRST require t∈B,THEN inspect EVERY A input for f(x)=t.This is inverse evaluation,not f(t)or1/f(t).

Step identifiers and dependencies

Rule: Inverse_EVALUATE_EACH_original_input_matches_typed_target · Uses: G001, S029

Reverse checks

  • S030 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S029 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S028 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S027 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S026 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S025 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S023 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S020 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S016 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S012 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

-3

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 3795944acc03dd20e6d07843565902e6ab6c5089ba9ce334c3667b6d489f57b1
  • candidate record SHA-256: e5847c334da551fc7eb6aa7369bdd685332d66d5a34386f12b4edb769eec2715
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON