# Typed Composition Paths and Slope-First Affine Orders

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261011-advance-20261015-r04-composition-foundations · 2026-10-11 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Complete Compatible Composition Paths and Range

First f, then g: Fin(0)=∅→Fin(0)→Fin(0); complete f=[], g=[]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([0, 0, 0], [], [])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([0, 0, 0], [], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0001/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 2. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 0, 3]], [3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 0, 3]], [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0002/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 3. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 0, 3]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 0, 3]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0003/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 4. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 0, 3]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 0, 3]], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0004/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 5. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 0, 3]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 0, 3]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0005/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 6. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 0, 3]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 0, 3]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0006/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 7. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 0, 3]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 0, 3]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0007/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 8. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 0, 3]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 0, 3]], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0008/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 9. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 0, 3]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 0, 3]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0009/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 10. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 0, 3]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 0, 3]], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0010/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 11. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 1, 1]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 1, 1]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0011/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 12. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 1, 1]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 1, 1]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0012/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 13. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 1, 1]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 1, 1]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0013/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 14. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 1, 1]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 1, 1]], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0014/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 15. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 1, 1]], [1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 1, 1]], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0015/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 16. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 1, 1]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 1, 1]], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0016/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 17. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 1, 1]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 1, 1]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0017/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 18. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 1, 1]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 1, 1]], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0018/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 19. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 1, 1]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 1, 1]], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0019/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 20. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 2, 0]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 2, 0]], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0020/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 21. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 2, 0]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 2, 0]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0021/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 22. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 2, 0]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 2, 0]], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0022/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 23. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 2, 0]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 2, 0]], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0023/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 24. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 2, 0]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 2, 0]], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0024/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 25. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 2, 0]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 2, 0]], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0025/)

Function composition · qf-density15-composition-compatible-path-table · candidate





## 26. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(0)=∅, middle=Fin(0)=∅, target=Fin(0)=∅. Complete f=[]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([0, 0, 0], [], [], [], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([0, 0, 0], [], [], [], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0026/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 27. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(3)={0,…,2}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 1, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([3, 3, 3], [0, 2], [0, 2], [2, 1], [1])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([3, 3, 3], [0, 2], [0, 2], [2, 1], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0027/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 28. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(3)={0,…,2}, middle=Fin(3)={0,…,2}, target=Fin(0)=∅. Complete f=[0, 1, 2]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([3, 3, 0], [], [], [], [0, 1, 2])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([3, 3, 0], [], [], [], [0, 1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0028/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 29. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0029/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 30. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0030/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 31. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0031/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 32. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0032/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 33. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0033/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 34. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0034/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 35. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0035/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 36. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 2, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0036/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 37. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0037/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 38. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0038/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 39. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0039/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 40. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0040/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 41. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0041/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 42. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0042/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 43. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0043/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 44. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 1, 2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0044/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 45. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 1], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0045/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 46. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 1], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0046/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 47. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 1], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0047/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 48. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 1], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0048/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 49. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 1], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0049/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 50. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 1], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0050/)

Function composition · qf-density15-composition-restricted-table-domain · candidate





## 51. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(1)x+(0), g(x)=(1)x+(0); ordered inputs=[0/1, 1/1, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([1, 0], [1, 0], [0/1, 1/1, -1/1], [0/1, 1/1, -1/1], 0)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([1, 0], [1, 0], [0/1, 1/1, -1/1], [0/1, 1/1, -1/1], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0051/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 52. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(1)x+(1), g(x)=(2)x+(0); ordered inputs=[0/1, 1/1, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([2, 2], [2, 1], [2/1, 4/1, 0/1], [1/1, 3/1, -1/1], 1)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([2, 2], [2, 1], [2/1, 4/1, 0/1], [1/1, 3/1, -1/1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0052/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 53. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(0)x+(1), g(x)=(0)x+(2); ordered inputs=[0/1, 1/1, 2/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([0, 2], [0, 1], [2/1, 2/1, 2/1], [1/1, 1/1, 1/1], 1)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([0, 2], [0, 1], [2/1, 2/1, 2/1], [1/1, 1/1, 1/1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0053/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 54. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(2)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-4, 3], [-4, 1], [3/1, 1/1, 7/1], [1/1, -1/1, 5/1], 2)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-4, 3], [-4, 1], [3/1, 1/1, 7/1], [1/1, -1/1, 5/1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0054/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 55. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(3)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-6, 1], [-6, 3], [1/1, -2/1, 7/1], [3/1, 0/1, 9/1], -2)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-6, 1], [-6, 3], [1/1, -2/1, 7/1], [3/1, 0/1, 9/1], -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0055/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 56. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(4)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-8, -1], [-8, 5], [-1/1, -5/1, 7/1], [5/1, 1/1, 13/1], -6)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-8, -1], [-8, 5], [-1/1, -5/1, 7/1], [5/1, 1/1, 13/1], -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0056/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 57. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(5)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-10, 3], [-10, 4], [3/1, -2/1, 13/1], [4/1, -1/1, 14/1], -1)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-10, 3], [-10, 4], [3/1, -2/1, 13/1], [4/1, -1/1, 14/1], -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0057/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 58. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(6)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-12, 1], [-12, 6], [1/1, -5/1, 13/1], [6/1, 0/1, 18/1], -5)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-12, 1], [-12, 6], [1/1, -5/1, 13/1], [6/1, 0/1, 18/1], -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0058/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 59. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(7)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-14, -1], [-14, 8], [-1/1, -8/1, 13/1], [8/1, 1/1, 22/1], -9)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-14, -1], [-14, 8], [-1/1, -8/1, 13/1], [8/1, 1/1, 22/1], -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0059/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 60. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(8)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-16, 3], [-16, 7], [3/1, -5/1, 19/1], [7/1, -1/1, 23/1], -4)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-16, 3], [-16, 7], [3/1, -5/1, 19/1], [7/1, -1/1, 23/1], -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0060/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 61. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(9)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0061/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 62. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(10)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-20, -1], [-20, 11], [-1/1, -11/1, 19/1], [11/1, 1/1, 31/1], -12)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-20, -1], [-20, 11], [-1/1, -11/1, 19/1], [11/1, 1/1, 31/1], -12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0062/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 63. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(11)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-22, 3], [-22, 10], [3/1, -8/1, 25/1], [10/1, -1/1, 32/1], -7)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-22, 3], [-22, 10], [3/1, -8/1, 25/1], [10/1, -1/1, 32/1], -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0063/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 64. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(12)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-24, 1], [-24, 12], [1/1, -11/1, 25/1], [12/1, 0/1, 36/1], -11)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-24, 1], [-24, 12], [1/1, -11/1, 25/1], [12/1, 0/1, 36/1], -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0064/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 65. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(13)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0065/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 66. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(14)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-28, 3], [-28, 13], [3/1, -11/1, 31/1], [13/1, -1/1, 41/1], -10)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-28, 3], [-28, 13], [3/1, -11/1, 31/1], [13/1, -1/1, 41/1], -10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0066/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 67. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(15)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-30, 1], [-30, 15], [1/1, -14/1, 31/1], [15/1, 0/1, 45/1], -14)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-30, 1], [-30, 15], [1/1, -14/1, 31/1], [15/1, 0/1, 45/1], -14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0067/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 68. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(16)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-32, -1], [-32, 17], [-1/1, -17/1, 31/1], [17/1, 1/1, 49/1], -18)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-32, -1], [-32, 17], [-1/1, -17/1, 31/1], [17/1, 1/1, 49/1], -18).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0068/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 69. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(17)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-34, 3], [-34, 16], [3/1, -14/1, 37/1], [16/1, -1/1, 50/1], -13)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-34, 3], [-34, 16], [3/1, -14/1, 37/1], [16/1, -1/1, 50/1], -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0069/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 70. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(18)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-36, 1], [-36, 18], [1/1, -17/1, 37/1], [18/1, 0/1, 54/1], -17)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-36, 1], [-36, 18], [1/1, -17/1, 37/1], [18/1, 0/1, 54/1], -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0070/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 71. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(19)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-38, -1], [-38, 20], [-1/1, -20/1, 37/1], [20/1, 1/1, 58/1], -21)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-38, -1], [-38, 20], [-1/1, -20/1, 37/1], [20/1, 1/1, 58/1], -21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0071/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 72. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(20)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-40, 3], [-40, 19], [3/1, -17/1, 43/1], [19/1, -1/1, 59/1], -16)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-40, 3], [-40, 19], [3/1, -17/1, 43/1], [19/1, -1/1, 59/1], -16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0072/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 73. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(21)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-42, 1], [-42, 21], [1/1, -20/1, 43/1], [21/1, 0/1, 63/1], -20)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-42, 1], [-42, 21], [1/1, -20/1, 43/1], [21/1, 0/1, 63/1], -20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0073/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 74. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(22)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-44, -1], [-44, 23], [-1/1, -23/1, 43/1], [23/1, 1/1, 67/1], -24)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-44, -1], [-44, 23], [-1/1, -23/1, 43/1], [23/1, 1/1, 67/1], -24).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0074/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 75. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(23)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0075/)

Function composition · qf-density15-composition-noncommuting-affine · candidate





## 76. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(0)=∅→Fin(0)→Fin(0)→Fin(0); complete f=[], g=[], h=[]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([0, 0, 0, 0], [], [], [], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([0, 0, 0, 0], [], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0076/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 77. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 0, 0], [0, 0, 0], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 0], [0, 0, 0], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0077/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 78. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 0, 0], [1, 0, 0], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 0], [1, 0, 0], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0078/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 79. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 0, 0], [3, 0, 0], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 0], [3, 0, 0], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0079/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 80. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 1, 0], [0, 1, 0], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 0], [0, 1, 0], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0080/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 81. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 1, 0], [1, 1, 0], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 0], [1, 1, 0], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0081/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 82. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 1, 0], [3, 1, 0], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 0], [3, 1, 0], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0082/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 83. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 3, 0], [0, 3, 0], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 3, 0], [0, 3, 0], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0083/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 84. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 3, 0], [1, 3, 0], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 3, 0], [1, 3, 0], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0084/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 85. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 3, 0], [3, 3, 0], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 3, 0], [3, 3, 0], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 0, 2, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0085/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 86. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 0, 1], [0, 0, 1], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 1], [0, 0, 1], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0086/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 87. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 0, 1], [1, 0, 1], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 1], [1, 0, 1], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0087/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 88. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 0, 1], [3, 0, 1], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 1], [3, 0, 1], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0088/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 89. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 1, 1], [0, 1, 1], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 1], [0, 1, 1], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0089/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 90. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 1, 1], [1, 1, 1], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 1], [1, 1, 1], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0090/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 91. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 1, 1], [3, 1, 1], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 1], [3, 1, 1], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0091/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 92. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 3, 1], [0, 3, 1], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 3, 1], [0, 3, 1], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0092/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 93. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 3, 1], [1, 3, 1], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 3, 1], [1, 3, 1], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0093/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 94. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 3, 1], [3, 3, 1], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 3, 1], [3, 3, 1], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 1, 0, 1]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0094/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 95. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 0, 3], [0, 0, 3], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 3], [0, 0, 3], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 2, 1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0095/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 96. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 0, 3], [1, 0, 3], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 3], [1, 0, 3], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 2, 1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0096/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 97. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 0, 3], [3, 0, 3], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 3], [3, 0, 3], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 2, 1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0097/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 98. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [0, 1, 3], [0, 1, 3], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 3], [0, 1, 3], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 2, 1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0098/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 99. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [1, 1, 3], [1, 1, 3], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 3], [1, 1, 3], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 2, 1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0099/)

Function composition · qf-density15-composition-three-map-parentheses · candidate





## 100. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation([3, 3, 3, 4], [3, 1, 3], [3, 1, 3], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 3], [3, 1, 3], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 2, 1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r04-composition-0100/)

Function composition · qf-density15-composition-three-map-parentheses · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r04-composition-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261011-advance-20261015-r04-composition-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r04-composition-foundations/validation-report.json)