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Inverse Function Foundations

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batch-20261004-advance-20261005-r04-inverse-function-foundations · 2026-10-04 · 100 questions

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

1. Inverse Table of a Finite Bijection

The declared total function f:A→B has A=∅, B=∅, and table ∅. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

∅

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as ∅. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

2. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-3, -2, -1}, B={1, 3, 5}, and table {(-3, 5), (-2, 1), (-1, 3)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -2), (3, -1), (5, -3)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -2), (3, -1), (5, -3)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

3. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-4, -3, -2, -1}, B={1, 3, 5, 7}, and table {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -3), (3, -2), (5, -1), (7, -4)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -3), (3, -2), (5, -1), (7, -4)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

4. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

5. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

6. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -6), (3, -5), (5, -4), (7, -3), (9, -2), (11, -1), (13, -7)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -6), (3, -5), (5, -4), (7, -3), (9, -2), (11, -1), (13, -7)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

7. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

8. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

9. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, and table {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

10. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, and table {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -10), (3, -9), (5, -8), (7, -7), (9, -6), (11, -5), (13, -4), (15, -3), (17, -2), (19, -1), (21, -11)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -10), (3, -9), (5, -8), (7, -7), (9, -6), (11, -5), (13, -4), (15, -3), (17, -2), (19, -1), (21, -11)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

11. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}, and table {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

12. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}, and table {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -12), (3, -11), (5, -10), (7, -9), (9, -8), (11, -7), (13, -6), (15, -5), (17, -4), (19, -3), (21, -2), (23, -1), (25, -13)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -12), (3, -11), (5, -10), (7, -9), (9, -8), (11, -7), (13, -6), (15, -5), (17, -4), (19, -3), (21, -2), (23, -1), (25, -13)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

13. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27}, and table {(-14, 27), (-13, 1), (-12, 3), (-11, 5), (-10, 7), (-9, 9), (-8, 11), (-7, 13), (-6, 15), (-5, 17), (-4, 19), (-3, 21), (-2, 23), (-1, 25)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -14)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -14)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

14. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}, and table {(-15, 29), (-14, 1), (-13, 3), (-12, 5), (-11, 7), (-10, 9), (-9, 11), (-8, 13), (-7, 15), (-6, 17), (-5, 19), (-4, 21), (-3, 23), (-2, 25), (-1, 27)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -14), (3, -13), (5, -12), (7, -11), (9, -10), (11, -9), (13, -8), (15, -7), (17, -6), (19, -5), (21, -4), (23, -3), (25, -2), (27, -1), (29, -15)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -14), (3, -13), (5, -12), (7, -11), (9, -10), (11, -9), (13, -8), (15, -7), (17, -6), (19, -5), (21, -4), (23, -3), (25, -2), (27, -1), (29, -15)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

15. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31}, and table {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

16. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}, and table {(-17, 33), (-16, 1), (-15, 3), (-14, 5), (-13, 7), (-12, 9), (-11, 11), (-10, 13), (-9, 15), (-8, 17), (-7, 19), (-6, 21), (-5, 23), (-4, 25), (-3, 27), (-2, 29), (-1, 31)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -16), (3, -15), (5, -14), (7, -13), (9, -12), (11, -11), (13, -10), (15, -9), (17, -8), (19, -7), (21, -6), (23, -5), (25, -4), (27, -3), (29, -2), (31, -1), (33, -17)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -16), (3, -15), (5, -14), (7, -13), (9, -12), (11, -11), (13, -10), (15, -9), (17, -8), (19, -7), (21, -6), (23, -5), (25, -4), (27, -3), (29, -2), (31, -1), (33, -17)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

17. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}, and table {(-18, 35), (-17, 1), (-16, 3), (-15, 5), (-14, 7), (-13, 9), (-12, 11), (-11, 13), (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23), (-5, 25), (-4, 27), (-3, 29), (-2, 31), (-1, 33)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -18)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -18)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

18. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37}, and table {(-19, 37), (-18, 1), (-17, 3), (-16, 5), (-15, 7), (-14, 9), (-13, 11), (-12, 13), (-11, 15), (-10, 17), (-9, 19), (-8, 21), (-7, 23), (-6, 25), (-5, 27), (-4, 29), (-3, 31), (-2, 33), (-1, 35)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -18), (3, -17), (5, -16), (7, -15), (9, -14), (11, -13), (13, -12), (15, -11), (17, -10), (19, -9), (21, -8), (23, -7), (25, -6), (27, -5), (29, -4), (31, -3), (33, -2), (35, -1), (37, -19)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -18), (3, -17), (5, -16), (7, -15), (9, -14), (11, -13), (13, -12), (15, -11), (17, -10), (19, -9), (21, -8), (23, -7), (25, -6), (27, -5), (29, -4), (31, -3), (33, -2), (35, -1), (37, -19)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

19. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39}, and table {(-20, 39), (-19, 1), (-18, 3), (-17, 5), (-16, 7), (-15, 9), (-14, 11), (-13, 13), (-12, 15), (-11, 17), (-10, 19), (-9, 21), (-8, 23), (-7, 25), (-6, 27), (-5, 29), (-4, 31), (-3, 33), (-2, 35), (-1, 37)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -20)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -20)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

20. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41}, and table {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -20), (3, -19), (5, -18), (7, -17), (9, -16), (11, -15), (13, -14), (15, -13), (17, -12), (19, -11), (21, -10), (23, -9), (25, -8), (27, -7), (29, -6), (31, -5), (33, -4), (35, -3), (37, -2), (39, -1), (41, -21)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -20), (3, -19), (5, -18), (7, -17), (9, -16), (11, -15), (13, -14), (15, -13), (17, -12), (19, -11), (21, -10), (23, -9), (25, -8), (27, -7), (29, -6), (31, -5), (33, -4), (35, -3), (37, -2), (39, -1), (41, -21)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

21. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43}, and table {(-22, 43), (-21, 1), (-20, 3), (-19, 5), (-18, 7), (-17, 9), (-16, 11), (-15, 13), (-14, 15), (-13, 17), (-12, 19), (-11, 21), (-10, 23), (-9, 25), (-8, 27), (-7, 29), (-6, 31), (-5, 33), (-4, 35), (-3, 37), (-2, 39), (-1, 41)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -22)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -22)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

22. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45}, and table {(-23, 45), (-22, 1), (-21, 3), (-20, 5), (-19, 7), (-18, 9), (-17, 11), (-16, 13), (-15, 15), (-14, 17), (-13, 19), (-12, 21), (-11, 23), (-10, 25), (-9, 27), (-8, 29), (-7, 31), (-6, 33), (-5, 35), (-4, 37), (-3, 39), (-2, 41), (-1, 43)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -22), (3, -21), (5, -20), (7, -19), (9, -18), (11, -17), (13, -16), (15, -15), (17, -14), (19, -13), (21, -12), (23, -11), (25, -10), (27, -9), (29, -8), (31, -7), (33, -6), (35, -5), (37, -4), (39, -3), (41, -2), (43, -1), (45, -23)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -22), (3, -21), (5, -20), (7, -19), (9, -18), (11, -17), (13, -16), (15, -15), (17, -14), (19, -13), (21, -12), (23, -11), (25, -10), (27, -9), (29, -8), (31, -7), (33, -6), (35, -5), (37, -4), (39, -3), (41, -2), (43, -1), (45, -23)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

23. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}, and table {(-24, 47), (-23, 1), (-22, 3), (-21, 5), (-20, 7), (-19, 9), (-18, 11), (-17, 13), (-16, 15), (-15, 17), (-14, 19), (-13, 21), (-12, 23), (-11, 25), (-10, 27), (-9, 29), (-8, 31), (-7, 33), (-6, 35), (-5, 37), (-4, 39), (-3, 41), (-2, 43), (-1, 45)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -24)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -24)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

24. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49}, and table {(-25, 49), (-24, 1), (-23, 3), (-22, 5), (-21, 7), (-20, 9), (-19, 11), (-18, 13), (-17, 15), (-16, 17), (-15, 19), (-14, 21), (-13, 23), (-12, 25), (-11, 27), (-10, 29), (-9, 31), (-8, 33), (-7, 35), (-6, 37), (-5, 39), (-4, 41), (-3, 43), (-2, 45), (-1, 47)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -24), (3, -23), (5, -22), (7, -21), (9, -20), (11, -19), (13, -18), (15, -17), (17, -16), (19, -15), (21, -14), (23, -13), (25, -12), (27, -11), (29, -10), (31, -9), (33, -8), (35, -7), (37, -6), (39, -5), (41, -4), (43, -3), (45, -2), (47, -1), (49, -25)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -24), (3, -23), (5, -22), (7, -21), (9, -20), (11, -19), (13, -18), (15, -17), (17, -16), (19, -15), (21, -14), (23, -13), (25, -12), (27, -11), (29, -10), (31, -9), (33, -8), (35, -7), (37, -6), (39, -5), (41, -4), (43, -3), (45, -2), (47, -1), (49, -25)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

25. Inverse Table of a Finite Bijection

The declared total function f:A→B has A={-26, -25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51}, and table {(-26, 51), (-25, 1), (-24, 3), (-23, 5), (-22, 7), (-21, 9), (-20, 11), (-19, 13), (-18, 15), (-17, 17), (-16, 19), (-15, 21), (-14, 23), (-13, 25), (-12, 27), (-11, 29), (-10, 31), (-9, 33), (-8, 35), (-7, 37), (-6, 39), (-5, 41), (-4, 43), (-3, 45), (-2, 47), (-1, 49)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

View answer and explanation

{(1, -25), (3, -24), (5, -23), (7, -22), (9, -21), (11, -20), (13, -19), (15, -18), (17, -17), (19, -16), (21, -15), (23, -14), (25, -13), (27, -12), (29, -11), (31, -10), (33, -9), (35, -8), (37, -7), (39, -6), (41, -5), (43, -4), (45, -3), (47, -2), (49, -1), (51, -26)}

Each target has a unique preimage, so exchanging the pairs gives the total inverse g:B→A as {(1, -25), (3, -24), (5, -23), (7, -22), (9, -21), (11, -20), (13, -19), (15, -18), (17, -17), (19, -16), (21, -15), (23, -14), (25, -13), (27, -12), (29, -11), (31, -10), (33, -9), (35, -8), (37, -7), (39, -6), (41, -5), (43, -4), (45, -3), (47, -2), (49, -1), (51, -26)}. The empty bijection has the empty inverse.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-table · candidate

26. Inverse Value at a Permitted Target

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

View answer and explanation

-1

Read the unique input mapped by f to 1, obtaining g(1)=-1. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

27. Inverse Value at a Permitted Target

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

View answer and explanation

-3

Read the unique input mapped by f to 5, obtaining g(5)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

28. Inverse Value at a Permitted Target

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

View answer and explanation

-3

Read the unique input mapped by f to 1, obtaining g(1)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

29. Inverse Value at a Permitted Target

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

View answer and explanation

-3

Read the unique input mapped by f to 3, obtaining g(3)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

30. Inverse Value at a Permitted Target

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

View answer and explanation

-3

Read the unique input mapped by f to 5, obtaining g(5)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

31. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 7∈B, find g(7).

View answer and explanation

-3

Read the unique input mapped by f to 7, obtaining g(7)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

32. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 9∈B, find g(9).

View answer and explanation

-3

Read the unique input mapped by f to 9, obtaining g(9)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

33. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 11∈B, find g(11).

View answer and explanation

-3

Read the unique input mapped by f to 11, obtaining g(11)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

34. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, and table {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 13∈B, find g(13).

View answer and explanation

-3

Read the unique input mapped by f to 13, obtaining g(13)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

35. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, and table {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 15∈B, find g(15).

View answer and explanation

-3

Read the unique input mapped by f to 15, obtaining g(15)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

36. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}, and table {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 17∈B, find g(17).

View answer and explanation

-3

Read the unique input mapped by f to 17, obtaining g(17)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

37. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}, and table {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 19∈B, find g(19).

View answer and explanation

-3

Read the unique input mapped by f to 19, obtaining g(19)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

38. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27}, and table {(-14, 27), (-13, 1), (-12, 3), (-11, 5), (-10, 7), (-9, 9), (-8, 11), (-7, 13), (-6, 15), (-5, 17), (-4, 19), (-3, 21), (-2, 23), (-1, 25)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 21∈B, find g(21).

View answer and explanation

-3

Read the unique input mapped by f to 21, obtaining g(21)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

39. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}, and table {(-15, 29), (-14, 1), (-13, 3), (-12, 5), (-11, 7), (-10, 9), (-9, 11), (-8, 13), (-7, 15), (-6, 17), (-5, 19), (-4, 21), (-3, 23), (-2, 25), (-1, 27)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 23∈B, find g(23).

View answer and explanation

-3

Read the unique input mapped by f to 23, obtaining g(23)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

40. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31}, and table {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 25∈B, find g(25).

View answer and explanation

-3

Read the unique input mapped by f to 25, obtaining g(25)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

41. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}, and table {(-17, 33), (-16, 1), (-15, 3), (-14, 5), (-13, 7), (-12, 9), (-11, 11), (-10, 13), (-9, 15), (-8, 17), (-7, 19), (-6, 21), (-5, 23), (-4, 25), (-3, 27), (-2, 29), (-1, 31)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 27∈B, find g(27).

View answer and explanation

-3

Read the unique input mapped by f to 27, obtaining g(27)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

42. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}, and table {(-18, 35), (-17, 1), (-16, 3), (-15, 5), (-14, 7), (-13, 9), (-12, 11), (-11, 13), (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23), (-5, 25), (-4, 27), (-3, 29), (-2, 31), (-1, 33)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 29∈B, find g(29).

View answer and explanation

-3

Read the unique input mapped by f to 29, obtaining g(29)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

43. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37}, and table {(-19, 37), (-18, 1), (-17, 3), (-16, 5), (-15, 7), (-14, 9), (-13, 11), (-12, 13), (-11, 15), (-10, 17), (-9, 19), (-8, 21), (-7, 23), (-6, 25), (-5, 27), (-4, 29), (-3, 31), (-2, 33), (-1, 35)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 31∈B, find g(31).

View answer and explanation

-3

Read the unique input mapped by f to 31, obtaining g(31)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

44. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39}, and table {(-20, 39), (-19, 1), (-18, 3), (-17, 5), (-16, 7), (-15, 9), (-14, 11), (-13, 13), (-12, 15), (-11, 17), (-10, 19), (-9, 21), (-8, 23), (-7, 25), (-6, 27), (-5, 29), (-4, 31), (-3, 33), (-2, 35), (-1, 37)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 33∈B, find g(33).

View answer and explanation

-3

Read the unique input mapped by f to 33, obtaining g(33)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

45. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41}, and table {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 35∈B, find g(35).

View answer and explanation

-3

Read the unique input mapped by f to 35, obtaining g(35)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

46. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43}, and table {(-22, 43), (-21, 1), (-20, 3), (-19, 5), (-18, 7), (-17, 9), (-16, 11), (-15, 13), (-14, 15), (-13, 17), (-12, 19), (-11, 21), (-10, 23), (-9, 25), (-8, 27), (-7, 29), (-6, 31), (-5, 33), (-4, 35), (-3, 37), (-2, 39), (-1, 41)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 37∈B, find g(37).

View answer and explanation

-3

Read the unique input mapped by f to 37, obtaining g(37)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

47. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45}, and table {(-23, 45), (-22, 1), (-21, 3), (-20, 5), (-19, 7), (-18, 9), (-17, 11), (-16, 13), (-15, 15), (-14, 17), (-13, 19), (-12, 21), (-11, 23), (-10, 25), (-9, 27), (-8, 29), (-7, 31), (-6, 33), (-5, 35), (-4, 37), (-3, 39), (-2, 41), (-1, 43)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 39∈B, find g(39).

View answer and explanation

-3

Read the unique input mapped by f to 39, obtaining g(39)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

48. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}, and table {(-24, 47), (-23, 1), (-22, 3), (-21, 5), (-20, 7), (-19, 9), (-18, 11), (-17, 13), (-16, 15), (-15, 17), (-14, 19), (-13, 21), (-12, 23), (-11, 25), (-10, 27), (-9, 29), (-8, 31), (-7, 33), (-6, 35), (-5, 37), (-4, 39), (-3, 41), (-2, 43), (-1, 45)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 41∈B, find g(41).

View answer and explanation

-3

Read the unique input mapped by f to 41, obtaining g(41)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

49. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49}, and table {(-25, 49), (-24, 1), (-23, 3), (-22, 5), (-21, 7), (-20, 9), (-19, 11), (-18, 13), (-17, 15), (-16, 17), (-15, 19), (-14, 21), (-13, 23), (-12, 25), (-11, 27), (-10, 29), (-9, 31), (-8, 33), (-7, 35), (-6, 37), (-5, 39), (-4, 41), (-3, 43), (-2, 45), (-1, 47)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 43∈B, find g(43).

View answer and explanation

-3

Read the unique input mapped by f to 43, obtaining g(43)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

50. Inverse Value at a Permitted Target

The declared total function f:A→B has A={-26, -25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51}, and table {(-26, 51), (-25, 1), (-24, 3), (-23, 5), (-22, 7), (-21, 9), (-20, 11), (-19, 13), (-18, 15), (-17, 17), (-16, 19), (-15, 21), (-14, 23), (-13, 25), (-12, 27), (-11, 29), (-10, 31), (-9, 33), (-8, 35), (-7, 37), (-6, 39), (-5, 41), (-4, 43), (-3, 45), (-2, 47), (-1, 49)}. The function has been confirmed bijective. For g=f⁻¹ and the permitted target 45∈B, find g(45).

View answer and explanation

-3

Read the unique input mapped by f to 45, obtaining g(45)=-3. Inverse evaluation is distinct from taking the reciprocal of a function value.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-evaluation · candidate

51. Checking Both Inverse Identities

The declared total function f:A→B has A=∅, B=∅, and table ∅. The function has been confirmed bijective. The candidate total g:B→A has table ∅. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

52. Checking Both Inverse Identities

The declared total function f:A→B has A={-3, -2, -1}, B={1, 3, 5}, and table {(-3, 5), (-2, 1), (-1, 3)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -1), (3, -3), (5, -2)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

53. Checking Both Inverse Identities

The declared total function f:A→B has A={-4, -3, -2, -1}, B={1, 3, 5, 7}, and table {(-4, 7), (-3, 1), (-2, 3), (-1, 5)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -3), (3, -2), (5, -1), (7, -4)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

54. Checking Both Inverse Identities

The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -3), (3, -2), (5, -1), (7, -5), (9, -4)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

55. Checking Both Inverse Identities

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11}, and table {(-6, 11), (-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -6)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

56. Checking Both Inverse Identities

The declared total function f:A→B has A={-7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-7, 13), (-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -5), (3, -4), (5, -3), (7, -2), (9, -1), (11, -7), (13, -6)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

57. Checking Both Inverse Identities

The declared total function f:A→B has A={-8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-8, 15), (-7, 1), (-6, 3), (-5, 5), (-4, 7), (-3, 9), (-2, 11), (-1, 13)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -8)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

58. Checking Both Inverse Identities

The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -7), (3, -6), (5, -5), (7, -4), (9, -3), (11, -2), (13, -1), (15, -9), (17, -8)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

59. Checking Both Inverse Identities

The declared total function f:A→B has A={-10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, and table {(-10, 19), (-9, 1), (-8, 3), (-7, 5), (-6, 7), (-5, 9), (-4, 11), (-3, 13), (-2, 15), (-1, 17)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -10)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

60. Checking Both Inverse Identities

The declared total function f:A→B has A={-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}, and table {(-11, 21), (-10, 1), (-9, 3), (-8, 5), (-7, 7), (-6, 9), (-5, 11), (-4, 13), (-3, 15), (-2, 17), (-1, 19)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -9), (3, -8), (5, -7), (7, -6), (9, -5), (11, -4), (13, -3), (15, -2), (17, -1), (19, -11), (21, -10)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

61. Checking Both Inverse Identities

The declared total function f:A→B has A={-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}, and table {(-12, 23), (-11, 1), (-10, 3), (-9, 5), (-8, 7), (-7, 9), (-6, 11), (-5, 13), (-4, 15), (-3, 17), (-2, 19), (-1, 21)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -12)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

62. Checking Both Inverse Identities

The declared total function f:A→B has A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}, and table {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -11), (3, -10), (5, -9), (7, -8), (9, -7), (11, -6), (13, -5), (15, -4), (17, -3), (19, -2), (21, -1), (23, -13), (25, -12)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

63. Checking Both Inverse Identities

The declared total function f:A→B has A={-14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27}, and table {(-14, 27), (-13, 1), (-12, 3), (-11, 5), (-10, 7), (-9, 9), (-8, 11), (-7, 13), (-6, 15), (-5, 17), (-4, 19), (-3, 21), (-2, 23), (-1, 25)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -14)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

64. Checking Both Inverse Identities

The declared total function f:A→B has A={-15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}, and table {(-15, 29), (-14, 1), (-13, 3), (-12, 5), (-11, 7), (-10, 9), (-9, 11), (-8, 13), (-7, 15), (-6, 17), (-5, 19), (-4, 21), (-3, 23), (-2, 25), (-1, 27)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -13), (3, -12), (5, -11), (7, -10), (9, -9), (11, -8), (13, -7), (15, -6), (17, -5), (19, -4), (21, -3), (23, -2), (25, -1), (27, -15), (29, -14)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

65. Checking Both Inverse Identities

The declared total function f:A→B has A={-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31}, and table {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

66. Checking Both Inverse Identities

The declared total function f:A→B has A={-17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}, and table {(-17, 33), (-16, 1), (-15, 3), (-14, 5), (-13, 7), (-12, 9), (-11, 11), (-10, 13), (-9, 15), (-8, 17), (-7, 19), (-6, 21), (-5, 23), (-4, 25), (-3, 27), (-2, 29), (-1, 31)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -17), (33, -16)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

67. Checking Both Inverse Identities

The declared total function f:A→B has A={-18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35}, and table {(-18, 35), (-17, 1), (-16, 3), (-15, 5), (-14, 7), (-13, 9), (-12, 11), (-11, 13), (-10, 15), (-9, 17), (-8, 19), (-7, 21), (-6, 23), (-5, 25), (-4, 27), (-3, 29), (-2, 31), (-1, 33)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -18)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

68. Checking Both Inverse Identities

The declared total function f:A→B has A={-19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37}, and table {(-19, 37), (-18, 1), (-17, 3), (-16, 5), (-15, 7), (-14, 9), (-13, 11), (-12, 13), (-11, 15), (-10, 17), (-9, 19), (-8, 21), (-7, 23), (-6, 25), (-5, 27), (-4, 29), (-3, 31), (-2, 33), (-1, 35)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -17), (3, -16), (5, -15), (7, -14), (9, -13), (11, -12), (13, -11), (15, -10), (17, -9), (19, -8), (21, -7), (23, -6), (25, -5), (27, -4), (29, -3), (31, -2), (33, -1), (35, -19), (37, -18)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

69. Checking Both Inverse Identities

The declared total function f:A→B has A={-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39}, and table {(-20, 39), (-19, 1), (-18, 3), (-17, 5), (-16, 7), (-15, 9), (-14, 11), (-13, 13), (-12, 15), (-11, 17), (-10, 19), (-9, 21), (-8, 23), (-7, 25), (-6, 27), (-5, 29), (-4, 31), (-3, 33), (-2, 35), (-1, 37)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -20)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

70. Checking Both Inverse Identities

The declared total function f:A→B has A={-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41}, and table {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -19), (3, -18), (5, -17), (7, -16), (9, -15), (11, -14), (13, -13), (15, -12), (17, -11), (19, -10), (21, -9), (23, -8), (25, -7), (27, -6), (29, -5), (31, -4), (33, -3), (35, -2), (37, -1), (39, -21), (41, -20)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

71. Checking Both Inverse Identities

The declared total function f:A→B has A={-22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43}, and table {(-22, 43), (-21, 1), (-20, 3), (-19, 5), (-18, 7), (-17, 9), (-16, 11), (-15, 13), (-14, 15), (-13, 17), (-12, 19), (-11, 21), (-10, 23), (-9, 25), (-8, 27), (-7, 29), (-6, 31), (-5, 33), (-4, 35), (-3, 37), (-2, 39), (-1, 41)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -22)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

72. Checking Both Inverse Identities

The declared total function f:A→B has A={-23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45}, and table {(-23, 45), (-22, 1), (-21, 3), (-20, 5), (-19, 7), (-18, 9), (-17, 11), (-16, 13), (-15, 15), (-14, 17), (-13, 19), (-12, 21), (-11, 23), (-10, 25), (-9, 27), (-8, 29), (-7, 31), (-6, 33), (-5, 35), (-4, 37), (-3, 39), (-2, 41), (-1, 43)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -21), (3, -20), (5, -19), (7, -18), (9, -17), (11, -16), (13, -15), (15, -14), (17, -13), (19, -12), (21, -11), (23, -10), (25, -9), (27, -8), (29, -7), (31, -6), (33, -5), (35, -4), (37, -3), (39, -2), (41, -1), (43, -23), (45, -22)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

73. Checking Both Inverse Identities

The declared total function f:A→B has A={-24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}, and table {(-24, 47), (-23, 1), (-22, 3), (-21, 5), (-20, 7), (-19, 9), (-18, 11), (-17, 13), (-16, 15), (-15, 17), (-14, 19), (-13, 21), (-12, 23), (-11, 25), (-10, 27), (-9, 29), (-8, 31), (-7, 33), (-6, 35), (-5, 37), (-4, 39), (-3, 41), (-2, 43), (-1, 45)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -24)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

74. Checking Both Inverse Identities

The declared total function f:A→B has A={-25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49}, and table {(-25, 49), (-24, 1), (-23, 3), (-22, 5), (-21, 7), (-20, 9), (-19, 11), (-18, 13), (-17, 15), (-16, 17), (-15, 19), (-14, 21), (-13, 23), (-12, 25), (-11, 27), (-10, 29), (-9, 31), (-8, 33), (-7, 35), (-6, 37), (-5, 39), (-4, 41), (-3, 43), (-2, 45), (-1, 47)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -23), (3, -22), (5, -21), (7, -20), (9, -19), (11, -18), (13, -17), (15, -16), (17, -15), (19, -14), (21, -13), (23, -12), (25, -11), (27, -10), (29, -9), (31, -8), (33, -7), (35, -6), (37, -5), (39, -4), (41, -3), (43, -2), (45, -1), (47, -25), (49, -24)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

No

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving No; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

75. Checking Both Inverse Identities

The declared total function f:A→B has A={-26, -25, -24, -23, -22, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51}, and table {(-26, 51), (-25, 1), (-24, 3), (-23, 5), (-22, 7), (-21, 9), (-20, 11), (-19, 13), (-18, 15), (-17, 17), (-16, 19), (-15, 21), (-14, 23), (-13, 25), (-12, 27), (-11, 29), (-10, 31), (-9, 33), (-8, 35), (-7, 37), (-6, 39), (-5, 41), (-4, 43), (-3, 45), (-2, 47), (-1, 49)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -25), (3, -24), (5, -23), (7, -22), (9, -21), (11, -20), (13, -19), (15, -18), (17, -17), (19, -16), (21, -15), (23, -14), (25, -13), (27, -12), (29, -11), (31, -10), (33, -9), (35, -8), (37, -7), (39, -6), (41, -5), (43, -4), (45, -3), (47, -2), (49, -1), (51, -26)}. Check both identities to decide whether g is the inverse of f.

View answer and explanation

Yes

Check g(f(x))=x for every x∈A and f(g(y))=y for every y∈B under the explicit types. Both restoration conditions must pass, giving Yes; omitted inputs cannot conceal failure.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-function-candidate · candidate

76. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(1)x+(-12), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/1, 12/1)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/1, 12/1), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

77. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-2)x+(-11), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/2, -11/2)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/2, -11/2), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

78. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(3)x+(-10), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/3, 10/3)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/3, 10/3), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

79. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-4)x+(-9), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/4, -9/4)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/4, -9/4), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

80. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(5)x+(-8), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/5, 8/5)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/5, 8/5), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

81. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-6)x+(-7), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/6, -7/6)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/6, -7/6), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

82. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(7)x+(-6), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/7, 6/7)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/7, 6/7), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

83. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-8)x+(-5), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/8, -5/8)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/8, -5/8), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

84. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(9)x+(-4), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/9, 4/9)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/9, 4/9), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

85. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-10)x+(-3), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/10, -3/10)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/10, -3/10), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

86. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(11)x+(-2), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/11, 2/11)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/11, 2/11), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

87. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-12)x+(-1), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/12, -1/12)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/12, -1/12), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

88. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(13)x+(0), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/13, 0/1)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/13, 0/1), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

89. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-14)x+(1), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/14, 1/14)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/14, 1/14), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

90. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(15)x+(2), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/15, -2/15)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/15, -2/15), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

91. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-16)x+(3), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/16, 3/16)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/16, 3/16), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

92. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(17)x+(4), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/17, -4/17)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/17, -4/17), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

93. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-18)x+(5), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/18, 5/18)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/18, 5/18), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

94. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(19)x+(6), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/19, -6/19)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/19, -6/19), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

95. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-20)x+(7), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/20, 7/20)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/20, 7/20), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

96. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(21)x+(8), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/21, -8/21)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/21, -8/21), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

97. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-22)x+(9), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/22, 9/22)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/22, 9/22), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

98. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(23)x+(10), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(1/23, -10/23)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/23, -10/23), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

99. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(-24)x+(11), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

View answer and explanation

(-1/24, 11/24)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (-1/24, 11/24), with nonzero slope ensuring invertibility in this real affine setting.

Reviewed step-by-step derivation

Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

100. Inverse Coefficients of a Nonzero-Slope Affine Function

Over R, let f:R→R be f(x)=(25)x+(12), with m≠0. Write the inverse as g(y)=uy+v and give (u,v) as reduced positive-denominator fractions, retaining /1 for integers.

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(1/25, -12/25)

Solving y=mx+b gives x=(y−b)/m, so u=1/m and v=−b/m. The exact coefficient pair is (1/25, -12/25), with nonzero slope ensuring invertibility in this real affine setting.

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Inverse functions of bijections · qf-inverse-affine-coefficients · candidate

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