Function Composition Foundations

-42

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

-6

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

4

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

-6

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

-30

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

43

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

24

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

-1

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

2

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

-3

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

0

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

-3

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

-6

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

10

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

-3

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

10

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

3

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

2

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

13

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

42

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

-65

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

16

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

-3

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

-4

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

19

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

-1

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

6

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

0

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

-7

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

12

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

0

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

-13

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

18

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

0

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

-19

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

24

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

0

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

-25

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

30

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

0

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

-31

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

36

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

0

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

-37

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

42

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

0

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

-43

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

48

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

0

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

-49

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

∅

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.

∅

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.

{-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.

∅

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.

{-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.

{-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.

{-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.

∅

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.

{-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.

{-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.

{-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.

∅

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.

{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.

{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.

{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.

∅

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.

{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.

{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.

{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.

∅

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.

{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.

{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.

{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.

∅

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.

{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.

Yes

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.

No

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.

No

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.

No

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.

No

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.

Yes

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.

No

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.

No

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.

No

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.

No

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.

Yes

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.

No

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.

No

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.

No

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.

No

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.

Yes

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.

No

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.

No

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.

Yes

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.

No

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.

Yes

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.

No

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.

No

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.

No

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.

No

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.