A={a,b,c,d}

X={1,2,3,4}

Each choice defines a function whose domain is A and whose target is X.

Select the function that has a well-defined inverse.

Question

A={a,b,c,d}

X={1,2,3,4}

Each choice defines a function whose domain is A and whose target is X.

Select the function that has a well-defined inverse.

BlackTom AI · Accepted Answer

A function has an inverse precisely when it is bijective: both injective (one-to-one) and surjective (onto). Only bijections produce a well-defined inverse function that assigns each element of the codomain a single preimage.

1) Relevant concepts
- Injective: distinct domain elements map to distinct codomain elements. If f(x1)=f(x2) with x1≠x2, f is not injective.
- Surjective: every element of the target set X is an image of some domain element.
- Inverse function f^{-1} exists (as a function) iff f is both injective and surjective.

2) Step-by-step check for each option (domain A={a,b,c,d}, target X={1,2,3,4})
Option 1: f={(a,3),(b,4),(c,3),(d,4)}
- Images: 3,4,3,4 → image set = {3,4}.
- Injectivity: fails, because a and c both map to 3 (and b and d both map to 4).
- Surjectivity: fails (1 and 2 are not hit).
- Conclusion: not bijective → no well-defined inverse.

Option 2: f={(a,3),(b,3),(c,3),(d,3)}
- Images: 3,3,3,3 → image set = {3}.
- Injectivity: fails (many domain elements share image 3).
- Surjectivity: fails (only 3 is hit).
- Conclusion: no inverse.

Option 3: f={(a,3),(b,4),(c,2),(d,1)}
- Images: 3,4,2,1 → image set = {1,2,3,4} = X.
- Injectivity: holds, since images are all distinct (no two domain elements share the same image).
- Surjectivity: holds, every element of X appears exactly once.
- Therefore f is bijective. The inverse is f^{-1} = {(3,a),(4,b),(2,c),(1,d)}, which is a well-defined function from X to A.

Option 4: f={(a,3),(b,4),(c,2),(d,4)}
- Images: 3,4,2,4 → image set = {2,3,4}.
- Injectivity: fails because b and d both map to 4.
- Surjectivity: fails (1 is not hit).
- Conclusion: not invertible.

3) Intermediate checks shown above (listing image sets and duplicates).

4) Why the correct answer is correct
Option 3 is the only mapping that is one-to-one and onto; each element of X corresponds to exactly one element of A, so the inverse exists and is a function.

5) Why other options are incorrect
They fail injectivity (and thus surjectivity), producing repeated images or missing elements of X, so their inverses would not be single-valued functions.

6) Conclusion
The correct choice is f = {(a,3),(b,4),(c,2),(d,1)}, matching the provided answer.

A={a,b,c,d} X={1,2,3,4} Each choice defines a function whose domain is A and whose target is X. Select the function that has a well-defined inverse.单项选择题

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