# Function Composition Foundations

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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-20260930-r04-composition-foundations · 2026-09-30 · 100 questions

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



## 1. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-3)t+(-12) and g:R→R be g(t)=(-2)t+(6). Find (g∘f)(-12) exactly.

View answer and explanation-42

First f(-12)=24; then g(f(-12))=-42.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0001/)

Function composition · qf-composition-affine-evaluation · candidate





## 2. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-2)t+(-11) and g:R→R be g(t)=(-1)t+(5). Find (g∘f)(-11) exactly.

View answer and explanation-6

First f(-11)=11; then g(f(-11))=-6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0002/)

Function composition · qf-composition-affine-evaluation · candidate





## 3. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-1)t+(-10) and g:R→R be g(t)=(0)t+(4). Find (g∘f)(-10) exactly.

View answer and explanation4

First f(-10)=0; then g(f(-10))=4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0003/)

Function composition · qf-composition-affine-evaluation · candidate





## 4. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(0)t+(-9) and g:R→R be g(t)=(1)t+(3). Find (g∘f)(-9) exactly.

View answer and explanation-6

First f(-9)=-9; then g(f(-9))=-6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0004/)

Function composition · qf-composition-affine-evaluation · candidate





## 5. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(1)t+(-8) and g:R→R be g(t)=(2)t+(2). Find (g∘f)(-8) exactly.

View answer and explanation-30

First f(-8)=-16; then g(f(-8))=-30.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0005/)

Function composition · qf-composition-affine-evaluation · candidate





## 6. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(2)t+(-7) and g:R→R be g(t)=(-2)t+(1). Find (g∘f)(-7) exactly.

View answer and explanation43

First f(-7)=-21; then g(f(-7))=43.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0006/)

Function composition · qf-composition-affine-evaluation · candidate





## 7. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(3)t+(-6) and g:R→R be g(t)=(-1)t+(0). Find (g∘f)(-6) exactly.

View answer and explanation24

First f(-6)=-24; then g(f(-6))=24.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0007/)

Function composition · qf-composition-affine-evaluation · candidate





## 8. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-3)t+(-5) and g:R→R be g(t)=(0)t+(-1). Find (g∘f)(-5) exactly.

View answer and explanation-1

First f(-5)=10; then g(f(-5))=-1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0008/)

Function composition · qf-composition-affine-evaluation · candidate





## 9. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-2)t+(-4) and g:R→R be g(t)=(1)t+(-2). Find (g∘f)(-4) exactly.

View answer and explanation2

First f(-4)=4; then g(f(-4))=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0009/)

Function composition · qf-composition-affine-evaluation · candidate





## 10. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-1)t+(-3) and g:R→R be g(t)=(2)t+(-3). Find (g∘f)(-3) exactly.

View answer and explanation-3

First f(-3)=0; then g(f(-3))=-3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0010/)

Function composition · qf-composition-affine-evaluation · candidate





## 11. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(0)t+(-2) and g:R→R be g(t)=(-2)t+(-4). Find (g∘f)(-2) exactly.

View answer and explanation0

First f(-2)=-2; then g(f(-2))=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0011/)

Function composition · qf-composition-affine-evaluation · candidate





## 12. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(1)t+(-1) and g:R→R be g(t)=(-1)t+(-5). Find (g∘f)(-1) exactly.

View answer and explanation-3

First f(-1)=-2; then g(f(-1))=-3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0012/)

Function composition · qf-composition-affine-evaluation · candidate





## 13. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(2)t+(0) and g:R→R be g(t)=(0)t+(-6). Find (g∘f)(0) exactly.

View answer and explanation-6

First f(0)=0; then g(f(0))=-6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0013/)

Function composition · qf-composition-affine-evaluation · candidate





## 14. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(3)t+(1) and g:R→R be g(t)=(1)t+(6). Find (g∘f)(1) exactly.

View answer and explanation10

First f(1)=4; then g(f(1))=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0014/)

Function composition · qf-composition-affine-evaluation · candidate





## 15. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-3)t+(2) and g:R→R be g(t)=(2)t+(5). Find (g∘f)(2) exactly.

View answer and explanation-3

First f(2)=-4; then g(f(2))=-3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0015/)

Function composition · qf-composition-affine-evaluation · candidate





## 16. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-2)t+(3) and g:R→R be g(t)=(-2)t+(4). Find (g∘f)(3) exactly.

View answer and explanation10

First f(3)=-3; then g(f(3))=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0016/)

Function composition · qf-composition-affine-evaluation · candidate





## 17. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-1)t+(4) and g:R→R be g(t)=(-1)t+(3). Find (g∘f)(4) exactly.

View answer and explanation3

First f(4)=0; then g(f(4))=3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0017/)

Function composition · qf-composition-affine-evaluation · candidate





## 18. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(0)t+(5) and g:R→R be g(t)=(0)t+(2). Find (g∘f)(5) exactly.

View answer and explanation2

First f(5)=5; then g(f(5))=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0018/)

Function composition · qf-composition-affine-evaluation · candidate





## 19. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(1)t+(6) and g:R→R be g(t)=(1)t+(1). Find (g∘f)(6) exactly.

View answer and explanation13

First f(6)=12; then g(f(6))=13.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0019/)

Function composition · qf-composition-affine-evaluation · candidate





## 20. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(2)t+(7) and g:R→R be g(t)=(2)t+(0). Find (g∘f)(7) exactly.

View answer and explanation42

First f(7)=21; then g(f(7))=42.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0020/)

Function composition · qf-composition-affine-evaluation · candidate





## 21. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(3)t+(8) and g:R→R be g(t)=(-2)t+(-1). Find (g∘f)(8) exactly.

View answer and explanation-65

First f(8)=32; then g(f(8))=-65.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0021/)

Function composition · qf-composition-affine-evaluation · candidate





## 22. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-3)t+(9) and g:R→R be g(t)=(-1)t+(-2). Find (g∘f)(9) exactly.

View answer and explanation16

First f(9)=-18; then g(f(9))=16.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0022/)

Function composition · qf-composition-affine-evaluation · candidate





## 23. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-2)t+(10) and g:R→R be g(t)=(0)t+(-3). Find (g∘f)(10) exactly.

View answer and explanation-3

First f(10)=-10; then g(f(10))=-3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0023/)

Function composition · qf-composition-affine-evaluation · candidate





## 24. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(-1)t+(11) and g:R→R be g(t)=(1)t+(-4). Find (g∘f)(11) exactly.

View answer and explanation-4

First f(11)=0; then g(f(11))=-4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0024/)

Function composition · qf-composition-affine-evaluation · candidate





## 25. Composition of Explicit Affine Expressions

Let f:R→R be f(t)=(0)t+(12) and g:R→R be g(t)=(2)t+(-5). Find (g∘f)(12) exactly.

View answer and explanation19

First f(12)=12; then g(f(12))=19.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-affine-evaluation-0025/)

Function composition · qf-composition-affine-evaluation · candidate





## 26. Composition Value from Finite Function Tables

The explicit total functions have A={-12, -11, -10}, B={-2, 0, 3}, C={-1, 0, 4}, f=[(-12, -2), (-11, 0), (-10, 3)], and g=[(-2, -1), (0, 4), (3, 0)]. Find (g∘f)(-12).

View answer and explanation-1

First read f(-12)=-2, then use that intermediate input in g to obtain -1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0001/)

Function composition · qf-composition-table-evaluation · candidate





## 27. Composition Value from Finite Function Tables

The explicit total functions have A={-11, -10, -9}, B={-3, 0, 4}, C={-3, 0, 6}, f=[(-11, 0), (-10, 4), (-9, -3)], and g=[(-3, 0), (0, -3), (4, 6)]. Find (g∘f)(-10).

View answer and explanation6

First read f(-10)=4, then use that intermediate input in g to obtain 6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0002/)

Function composition · qf-composition-table-evaluation · candidate





## 28. Composition Value from Finite Function Tables

The explicit total functions have A={-10, -9, -8}, B={-4, 0, 5}, C={-5, 0, 8}, f=[(-10, 5), (-9, -4), (-8, 0)], and g=[(-4, 8), (0, 0), (5, -5)]. Find (g∘f)(-8).

View answer and explanation0

First read f(-8)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0003/)

Function composition · qf-composition-table-evaluation · candidate





## 29. Composition Value from Finite Function Tables

The explicit total functions have A={-9, -8, -7}, B={-5, 0, 6}, C={-7, 0, 10}, f=[(-9, -5), (-8, 0), (-7, 6)], and g=[(-5, -7), (0, 10), (6, 0)]. Find (g∘f)(-9).

View answer and explanation-7

First read f(-9)=-5, then use that intermediate input in g to obtain -7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0004/)

Function composition · qf-composition-table-evaluation · candidate





## 30. Composition Value from Finite Function Tables

The explicit total functions have A={-8, -7, -6}, B={-6, 0, 7}, C={-9, 0, 12}, f=[(-8, 0), (-7, 7), (-6, -6)], and g=[(-6, 0), (0, -9), (7, 12)]. Find (g∘f)(-7).

View answer and explanation12

First read f(-7)=7, then use that intermediate input in g to obtain 12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0005/)

Function composition · qf-composition-table-evaluation · candidate





## 31. Composition Value from Finite Function Tables

The explicit total functions have A={-7, -6, -5}, B={-7, 0, 8}, C={-11, 0, 14}, f=[(-7, 8), (-6, -7), (-5, 0)], and g=[(-7, 14), (0, 0), (8, -11)]. Find (g∘f)(-5).

View answer and explanation0

First read f(-5)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0006/)

Function composition · qf-composition-table-evaluation · candidate





## 32. Composition Value from Finite Function Tables

The explicit total functions have A={-6, -5, -4}, B={-8, 0, 9}, C={-13, 0, 16}, f=[(-6, -8), (-5, 0), (-4, 9)], and g=[(-8, -13), (0, 16), (9, 0)]. Find (g∘f)(-6).

View answer and explanation-13

First read f(-6)=-8, then use that intermediate input in g to obtain -13.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0007/)

Function composition · qf-composition-table-evaluation · candidate





## 33. Composition Value from Finite Function Tables

The explicit total functions have A={-5, -4, -3}, B={-9, 0, 10}, C={-15, 0, 18}, f=[(-5, 0), (-4, 10), (-3, -9)], and g=[(-9, 0), (0, -15), (10, 18)]. Find (g∘f)(-4).

View answer and explanation18

First read f(-4)=10, then use that intermediate input in g to obtain 18.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0008/)

Function composition · qf-composition-table-evaluation · candidate





## 34. Composition Value from Finite Function Tables

The explicit total functions have A={-4, -3, -2}, B={-10, 0, 11}, C={-17, 0, 20}, f=[(-4, 11), (-3, -10), (-2, 0)], and g=[(-10, 20), (0, 0), (11, -17)]. Find (g∘f)(-2).

View answer and explanation0

First read f(-2)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0009/)

Function composition · qf-composition-table-evaluation · candidate





## 35. Composition Value from Finite Function Tables

The explicit total functions have A={-3, -2, -1}, B={-11, 0, 12}, C={-19, 0, 22}, f=[(-3, -11), (-2, 0), (-1, 12)], and g=[(-11, -19), (0, 22), (12, 0)]. Find (g∘f)(-3).

View answer and explanation-19

First read f(-3)=-11, then use that intermediate input in g to obtain -19.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0010/)

Function composition · qf-composition-table-evaluation · candidate





## 36. Composition Value from Finite Function Tables

The explicit total functions have A={-2, -1, 0}, B={-12, 0, 13}, C={-21, 0, 24}, f=[(-2, 0), (-1, 13), (0, -12)], and g=[(-12, 0), (0, -21), (13, 24)]. Find (g∘f)(-1).

View answer and explanation24

First read f(-1)=13, then use that intermediate input in g to obtain 24.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0011/)

Function composition · qf-composition-table-evaluation · candidate





## 37. Composition Value from Finite Function Tables

The explicit total functions have A={-1, 0, 1}, B={-13, 0, 14}, C={-23, 0, 26}, f=[(-1, 14), (0, -13), (1, 0)], and g=[(-13, 26), (0, 0), (14, -23)]. Find (g∘f)(1).

View answer and explanation0

First read f(1)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0012/)

Function composition · qf-composition-table-evaluation · candidate





## 38. Composition Value from Finite Function Tables

The explicit total functions have A={0, 1, 2}, B={-14, 0, 15}, C={-25, 0, 28}, f=[(0, -14), (1, 0), (2, 15)], and g=[(-14, -25), (0, 28), (15, 0)]. Find (g∘f)(0).

View answer and explanation-25

First read f(0)=-14, then use that intermediate input in g to obtain -25.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0013/)

Function composition · qf-composition-table-evaluation · candidate





## 39. Composition Value from Finite Function Tables

The explicit total functions have A={1, 2, 3}, B={-15, 0, 16}, C={-27, 0, 30}, f=[(1, 0), (2, 16), (3, -15)], and g=[(-15, 0), (0, -27), (16, 30)]. Find (g∘f)(2).

View answer and explanation30

First read f(2)=16, then use that intermediate input in g to obtain 30.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0014/)

Function composition · qf-composition-table-evaluation · candidate





## 40. Composition Value from Finite Function Tables

The explicit total functions have A={2, 3, 4}, B={-16, 0, 17}, C={-29, 0, 32}, f=[(2, 17), (3, -16), (4, 0)], and g=[(-16, 32), (0, 0), (17, -29)]. Find (g∘f)(4).

View answer and explanation0

First read f(4)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0015/)

Function composition · qf-composition-table-evaluation · candidate





## 41. Composition Value from Finite Function Tables

The explicit total functions have A={3, 4, 5}, B={-17, 0, 18}, C={-31, 0, 34}, f=[(3, -17), (4, 0), (5, 18)], and g=[(-17, -31), (0, 34), (18, 0)]. Find (g∘f)(3).

View answer and explanation-31

First read f(3)=-17, then use that intermediate input in g to obtain -31.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0016/)

Function composition · qf-composition-table-evaluation · candidate





## 42. Composition Value from Finite Function Tables

The explicit total functions have A={4, 5, 6}, B={-18, 0, 19}, C={-33, 0, 36}, f=[(4, 0), (5, 19), (6, -18)], and g=[(-18, 0), (0, -33), (19, 36)]. Find (g∘f)(5).

View answer and explanation36

First read f(5)=19, then use that intermediate input in g to obtain 36.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0017/)

Function composition · qf-composition-table-evaluation · candidate





## 43. Composition Value from Finite Function Tables

The explicit total functions have A={5, 6, 7}, B={-19, 0, 20}, C={-35, 0, 38}, f=[(5, 20), (6, -19), (7, 0)], and g=[(-19, 38), (0, 0), (20, -35)]. Find (g∘f)(7).

View answer and explanation0

First read f(7)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0018/)

Function composition · qf-composition-table-evaluation · candidate





## 44. Composition Value from Finite Function Tables

The explicit total functions have A={6, 7, 8}, B={-20, 0, 21}, C={-37, 0, 40}, f=[(6, -20), (7, 0), (8, 21)], and g=[(-20, -37), (0, 40), (21, 0)]. Find (g∘f)(6).

View answer and explanation-37

First read f(6)=-20, then use that intermediate input in g to obtain -37.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0019/)

Function composition · qf-composition-table-evaluation · candidate





## 45. Composition Value from Finite Function Tables

The explicit total functions have A={7, 8, 9}, B={-21, 0, 22}, C={-39, 0, 42}, f=[(7, 0), (8, 22), (9, -21)], and g=[(-21, 0), (0, -39), (22, 42)]. Find (g∘f)(8).

View answer and explanation42

First read f(8)=22, then use that intermediate input in g to obtain 42.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0020/)

Function composition · qf-composition-table-evaluation · candidate





## 46. Composition Value from Finite Function Tables

The explicit total functions have A={8, 9, 10}, B={-22, 0, 23}, C={-41, 0, 44}, f=[(8, 23), (9, -22), (10, 0)], and g=[(-22, 44), (0, 0), (23, -41)]. Find (g∘f)(10).

View answer and explanation0

First read f(10)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0021/)

Function composition · qf-composition-table-evaluation · candidate





## 47. Composition Value from Finite Function Tables

The explicit total functions have A={9, 10, 11}, B={-23, 0, 24}, C={-43, 0, 46}, f=[(9, -23), (10, 0), (11, 24)], and g=[(-23, -43), (0, 46), (24, 0)]. Find (g∘f)(9).

View answer and explanation-43

First read f(9)=-23, then use that intermediate input in g to obtain -43.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0022/)

Function composition · qf-composition-table-evaluation · candidate





## 48. Composition Value from Finite Function Tables

The explicit total functions have A={10, 11, 12}, B={-24, 0, 25}, C={-45, 0, 48}, f=[(10, 0), (11, 25), (12, -24)], and g=[(-24, 0), (0, -45), (25, 48)]. Find (g∘f)(11).

View answer and explanation48

First read f(11)=25, then use that intermediate input in g to obtain 48.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0023/)

Function composition · qf-composition-table-evaluation · candidate





## 49. Composition Value from Finite Function Tables

The explicit total functions have A={11, 12, 13}, B={-25, 0, 26}, C={-47, 0, 50}, f=[(11, 26), (12, -25), (13, 0)], and g=[(-25, 50), (0, 0), (26, -47)]. Find (g∘f)(13).

View answer and explanation0

First read f(13)=0, then use that intermediate input in g to obtain 0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0024/)

Function composition · qf-composition-table-evaluation · candidate





## 50. Composition Value from Finite Function Tables

The explicit total functions have A={12, 13, 14}, B={-26, 0, 27}, C={-49, 0, 52}, f=[(12, -26), (13, 0), (14, 27)], and g=[(-26, -49), (0, 52), (27, 0)]. Find (g∘f)(12).

View answer and explanation-49

First read f(12)=-26, then use that intermediate input in g to obtain -49.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-table-evaluation-0025/)

Function composition · qf-composition-table-evaluation · candidate





## 51. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A=∅, B={0}, D=∅, C=∅, f=[], and g=[]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0001/)

Function composition · qf-composition-domain-set · candidate





## 52. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-1, 0, 1}, B={-2, 0, 2}, D=∅, C={5}, f=[(-1, -2), (0, 0), (1, 2)], and g=[]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0002/)

Function composition · qf-composition-domain-set · candidate





## 53. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-10, -9, -8}, B={-4, 0, 5}, D={-4, 0}, C={-4, 1, 5}, f=[(-10, 5), (-9, -4), (-8, 0)], and g=[(-4, 5), (0, -4)]. List {x∈A:f(x)∈D}.

View answer and explanation{-9, -8}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-9, -8}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0003/)

Function composition · qf-composition-domain-set · candidate





## 54. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-9, -8, -7}, B={-5, 0, 6}, D={102}, C={-6, 1, 7}, f=[(-9, -5), (-8, 0), (-7, 6)], and g=[(102, -6)]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0004/)

Function composition · qf-composition-domain-set · candidate





## 55. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-8, -7, -6}, B={-6, 0, 7}, D={-6, 0, 7}, C={-8, 1, 9}, f=[(-8, 0), (-7, 7), (-6, -6)], and g=[(-6, 1), (0, 9), (7, -8)]. List {x∈A:f(x)∈D}.

View answer and explanation{-8, -7, -6}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-8, -7, -6}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0005/)

Function composition · qf-composition-domain-set · candidate





## 56. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-7, -6, -5}, B={-7, 0, 8}, D={0}, C={-10, 1, 11}, f=[(-7, 8), (-6, -7), (-5, 0)], and g=[(0, 11)]. List {x∈A:f(x)∈D}.

View answer and explanation{-5}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-5}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0006/)

Function composition · qf-composition-domain-set · candidate





## 57. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-6, -5, -4}, B={-8, 0, 9}, D={-8, 0}, C={-12, 1, 13}, f=[(-6, -8), (-5, 0), (-4, 9)], and g=[(-8, -12), (0, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation{-6, -5}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-6, -5}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0007/)

Function composition · qf-composition-domain-set · candidate





## 58. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-5, -4, -3}, B={-9, 0, 10}, D={106}, C={-14, 1, 15}, f=[(-5, 0), (-4, 10), (-3, -9)], and g=[(106, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0008/)

Function composition · qf-composition-domain-set · candidate





## 59. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-4, -3, -2}, B={-10, 0, 11}, D={-10, 0, 11}, C={-16, 1, 17}, f=[(-4, 11), (-3, -10), (-2, 0)], and g=[(-10, 17), (0, -16), (11, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation{-4, -3, -2}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-4, -3, -2}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0009/)

Function composition · qf-composition-domain-set · candidate





## 60. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-3, -2, -1}, B={-11, 0, 12}, D={0}, C={-18, 1, 19}, f=[(-3, -11), (-2, 0), (-1, 12)], and g=[(0, -18)]. List {x∈A:f(x)∈D}.

View answer and explanation{-2}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-2}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0010/)

Function composition · qf-composition-domain-set · candidate





## 61. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-2, -1, 0}, B={-12, 0, 13}, D={-12, 0}, C={-20, 1, 21}, f=[(-2, 0), (-1, 13), (0, -12)], and g=[(-12, 1), (0, 21)]. List {x∈A:f(x)∈D}.

View answer and explanation{-2, 0}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {-2, 0}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0011/)

Function composition · qf-composition-domain-set · candidate





## 62. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={-1, 0, 1}, B={-13, 0, 14}, D={110}, C={-22, 1, 23}, f=[(-1, 14), (0, -13), (1, 0)], and g=[(110, 23)]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0012/)

Function composition · qf-composition-domain-set · candidate





## 63. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={0, 1, 2}, B={-14, 0, 15}, D={-14, 0, 15}, C={-24, 1, 25}, f=[(0, -14), (1, 0), (2, 15)], and g=[(-14, -24), (0, 1), (15, 25)]. List {x∈A:f(x)∈D}.

View answer and explanation{0, 1, 2}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {0, 1, 2}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0013/)

Function composition · qf-composition-domain-set · candidate





## 64. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={1, 2, 3}, B={-15, 0, 16}, D={0}, C={-26, 1, 27}, f=[(1, 0), (2, 16), (3, -15)], and g=[(0, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation{1}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {1}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0014/)

Function composition · qf-composition-domain-set · candidate





## 65. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={2, 3, 4}, B={-16, 0, 17}, D={-16, 0}, C={-28, 1, 29}, f=[(2, 17), (3, -16), (4, 0)], and g=[(-16, 29), (0, -28)]. List {x∈A:f(x)∈D}.

View answer and explanation{3, 4}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {3, 4}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0015/)

Function composition · qf-composition-domain-set · candidate





## 66. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={3, 4, 5}, B={-17, 0, 18}, D={114}, C={-30, 1, 31}, f=[(3, -17), (4, 0), (5, 18)], and g=[(114, -30)]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0016/)

Function composition · qf-composition-domain-set · candidate





## 67. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={4, 5, 6}, B={-18, 0, 19}, D={-18, 0, 19}, C={-32, 1, 33}, f=[(4, 0), (5, 19), (6, -18)], and g=[(-18, 1), (0, 33), (19, -32)]. List {x∈A:f(x)∈D}.

View answer and explanation{4, 5, 6}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {4, 5, 6}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0017/)

Function composition · qf-composition-domain-set · candidate





## 68. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={5, 6, 7}, B={-19, 0, 20}, D={0}, C={-34, 1, 35}, f=[(5, 20), (6, -19), (7, 0)], and g=[(0, 35)]. List {x∈A:f(x)∈D}.

View answer and explanation{7}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {7}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0018/)

Function composition · qf-composition-domain-set · candidate





## 69. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={6, 7, 8}, B={-20, 0, 21}, D={-20, 0}, C={-36, 1, 37}, f=[(6, -20), (7, 0), (8, 21)], and g=[(-20, -36), (0, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation{6, 7}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {6, 7}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0019/)

Function composition · qf-composition-domain-set · candidate





## 70. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={7, 8, 9}, B={-21, 0, 22}, D={118}, C={-38, 1, 39}, f=[(7, 0), (8, 22), (9, -21)], and g=[(118, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0020/)

Function composition · qf-composition-domain-set · candidate





## 71. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={8, 9, 10}, B={-22, 0, 23}, D={-22, 0, 23}, C={-40, 1, 41}, f=[(8, 23), (9, -22), (10, 0)], and g=[(-22, 41), (0, -40), (23, 1)]. List {x∈A:f(x)∈D}.

View answer and explanation{8, 9, 10}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {8, 9, 10}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0021/)

Function composition · qf-composition-domain-set · candidate





## 72. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={9, 10, 11}, B={-23, 0, 24}, D={0}, C={-42, 1, 43}, f=[(9, -23), (10, 0), (11, 24)], and g=[(0, -42)]. List {x∈A:f(x)∈D}.

View answer and explanation{10}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {10}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0022/)

Function composition · qf-composition-domain-set · candidate





## 73. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={10, 11, 12}, B={-24, 0, 25}, D={-24, 0}, C={-44, 1, 45}, f=[(10, 0), (11, 25), (12, -24)], and g=[(-24, 1), (0, 45)]. List {x∈A:f(x)∈D}.

View answer and explanation{10, 12}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {10, 12}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0023/)

Function composition · qf-composition-domain-set · candidate





## 74. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={11, 12, 13}, B={-25, 0, 26}, D={122}, C={-46, 1, 47}, f=[(11, 26), (12, -25), (13, 0)], and g=[(122, 47)]. List {x∈A:f(x)∈D}.

View answer and explanation∅

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is ∅; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0024/)

Function composition · qf-composition-domain-set · candidate





## 75. Permitted Domain of a Restricted Composition

Let f:A→B and g:D→C be the explicit total functions A={12, 13, 14}, B={-26, 0, 27}, D={-26, 0, 27}, C={-48, 1, 49}, f=[(12, -26), (13, 0), (14, 27)], and g=[(-26, -48), (0, 1), (27, 49)]. List {x∈A:f(x)∈D}.

View answer and explanation{12, 13, 14}

Check each input in A and retain it exactly when its f-value lies in D. The permitted composition domain is {12, 13, 14}; it may be empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-domain-set-0025/)

Function composition · qf-composition-domain-set · candidate





## 76. Comparing Two Composition Orders

Let f(t)=(1)t+(-12) and g(t)=(1)t+(6). At the explicit input x=-12, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationYes

The two values are -18 and -18, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0001/)

Function composition · qf-composition-order-comparison · candidate





## 77. Comparing Two Composition Orders

Let f(t)=(-2)t+(-11) and g(t)=(-1)t+(5). At the explicit input x=-11, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -6 and -43, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0002/)

Function composition · qf-composition-order-comparison · candidate





## 78. Comparing Two Composition Orders

Let f(t)=(-1)t+(-10) and g(t)=(0)t+(4). At the explicit input x=-10, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 4 and -14, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0003/)

Function composition · qf-composition-order-comparison · candidate





## 79. Comparing Two Composition Orders

Let f(t)=(0)t+(-9) and g(t)=(1)t+(3). At the explicit input x=-9, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -6 and -9, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0004/)

Function composition · qf-composition-order-comparison · candidate





## 80. Comparing Two Composition Orders

Let f(t)=(1)t+(-8) and g(t)=(2)t+(2). At the explicit input x=-8, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -30 and -22, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0005/)

Function composition · qf-composition-order-comparison · candidate





## 81. Comparing Two Composition Orders

Let f(t)=(1)t+(-7) and g(t)=(1)t+(1). At the explicit input x=-7, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationYes

The two values are -13 and -13, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0006/)

Function composition · qf-composition-order-comparison · candidate





## 82. Comparing Two Composition Orders

Let f(t)=(3)t+(-6) and g(t)=(-1)t+(0). At the explicit input x=-6, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 24 and 12, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0007/)

Function composition · qf-composition-order-comparison · candidate





## 83. Comparing Two Composition Orders

Let f(t)=(-3)t+(-5) and g(t)=(0)t+(-1). At the explicit input x=-5, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -1 and -2, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0008/)

Function composition · qf-composition-order-comparison · candidate





## 84. Comparing Two Composition Orders

Let f(t)=(-2)t+(-4) and g(t)=(1)t+(-2). At the explicit input x=-4, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 2 and 8, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0009/)

Function composition · qf-composition-order-comparison · candidate





## 85. Comparing Two Composition Orders

Let f(t)=(-1)t+(-3) and g(t)=(2)t+(-3). At the explicit input x=-3, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -3 and 6, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0010/)

Function composition · qf-composition-order-comparison · candidate





## 86. Comparing Two Composition Orders

Let f(t)=(1)t+(-2) and g(t)=(1)t+(-4). At the explicit input x=-2, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationYes

The two values are -8 and -8, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0011/)

Function composition · qf-composition-order-comparison · candidate





## 87. Comparing Two Composition Orders

Let f(t)=(1)t+(-1) and g(t)=(-1)t+(-5). At the explicit input x=-1, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -3 and -5, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0012/)

Function composition · qf-composition-order-comparison · candidate





## 88. Comparing Two Composition Orders

Let f(t)=(2)t+(0) and g(t)=(0)t+(-6). At the explicit input x=0, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -6 and -12, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0013/)

Function composition · qf-composition-order-comparison · candidate





## 89. Comparing Two Composition Orders

Let f(t)=(3)t+(1) and g(t)=(1)t+(6). At the explicit input x=1, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 10 and 22, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0014/)

Function composition · qf-composition-order-comparison · candidate





## 90. Comparing Two Composition Orders

Let f(t)=(-3)t+(2) and g(t)=(2)t+(5). At the explicit input x=2, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -3 and -25, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0015/)

Function composition · qf-composition-order-comparison · candidate





## 91. Comparing Two Composition Orders

Let f(t)=(1)t+(3) and g(t)=(1)t+(4). At the explicit input x=3, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationYes

The two values are 10 and 10, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0016/)

Function composition · qf-composition-order-comparison · candidate





## 92. Comparing Two Composition Orders

Let f(t)=(-1)t+(4) and g(t)=(-1)t+(3). At the explicit input x=4, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 3 and 5, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0017/)

Function composition · qf-composition-order-comparison · candidate





## 93. Comparing Two Composition Orders

Let f(t)=(0)t+(5) and g(t)=(0)t+(2). At the explicit input x=5, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 2 and 5, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0018/)

Function composition · qf-composition-order-comparison · candidate





## 94. Comparing Two Composition Orders

Let f(t)=(1)t+(6) and g(t)=(1)t+(1). At the explicit input x=6, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationYes

The two values are 13 and 13, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0019/)

Function composition · qf-composition-order-comparison · candidate





## 95. Comparing Two Composition Orders

Let f(t)=(2)t+(7) and g(t)=(2)t+(0). At the explicit input x=7, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 42 and 35, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0020/)

Function composition · qf-composition-order-comparison · candidate





## 96. Comparing Two Composition Orders

Let f(t)=(1)t+(8) and g(t)=(1)t+(-1). At the explicit input x=8, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationYes

The two values are 15 and 15, so the answer is Yes. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0021/)

Function composition · qf-composition-order-comparison · candidate





## 97. Comparing Two Composition Orders

Let f(t)=(-3)t+(9) and g(t)=(-1)t+(-2). At the explicit input x=9, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 16 and 42, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0022/)

Function composition · qf-composition-order-comparison · candidate





## 98. Comparing Two Composition Orders

Let f(t)=(-2)t+(10) and g(t)=(0)t+(-3). At the explicit input x=10, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -3 and 16, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0023/)

Function composition · qf-composition-order-comparison · candidate





## 99. Comparing Two Composition Orders

Let f(t)=(-1)t+(11) and g(t)=(1)t+(-4). At the explicit input x=11, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are -4 and 4, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0024/)

Function composition · qf-composition-order-comparison · candidate





## 100. Comparing Two Composition Orders

Let f(t)=(0)t+(12) and g(t)=(2)t+(-5). At the explicit input x=12, are (g∘f)(x) and (f∘g)(x) equal?

View answer and explanationNo

The two values are 19 and 12, so the answer is No. Equality or inequality at this one input does not prove a general commutativity claim.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r04-composition-order-comparison-0025/)

Function composition · qf-composition-order-comparison · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260930-r04-composition-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260930-r04-composition-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260930-r04-composition-foundations/validation-report.json)