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

a) f={a, 3), (b, 4), }
b) f={(c, 2), (d, 1)}
c) f={a. 3), (b, 4), }
d) f={(c, 2), (d, 4))

Answer 1

The function that has a well-defined inverse is f={(c, 2), (d, 1)}, indicating option b) is correct. This is because the function is bijective, as it is both one-to-one and onto.

Step-by-step explanation:

The student is asking about functions with well-defined inverses. A function has a well-defined inverse if and only if it is bijective, which means the function is both injective (or one-to-one) and surjective (onto). Looking at the choices provided, let's evaluate each:

• f={a, 3), (b, 4), (b, 1)} - This function is not one-to-one as the element 'b' maps to two different elements in the range, hence it cannot have an inverse.
• f={(c, 2), (d, 1)} - This function is one-to-one and onto, as each element of A is paired with a unique element of X and every element of X is used. Therefore, this function has a well-defined inverse.
• f={a, 3), (b, 4), (c, 1)} - While this function appears to be one-to-one, it is not surjective since '2' in the range X is not used, making the inverse not well-defined.
• f={(c, 2), (d, 4)} - Similar to the first case, this function is not one-to-one because both 'c' and 'd' map to '2,' hence it cannot have a well-defined inverse.

Therefore, the correct choice for the function that has a well-defined inverse is f={(c, 2), (d, 1)}, which is option b).