Invertible Functions

A function is said to be invertible if it has an inverse function. In other words, an invertible function can be reversed or "undone" by another function.

Bijective Functions

A function is bijective if it is both injective (one-to-one) and surjective (onto).

Injective Functions

An injective function, also known as a one-to-one function, is a function where each element in the domain is mapped to a unique element in the codomain. In other words, no two distinct elements in the domain are mapped to the same element in the codomain.

Surjective Functions

A surjective function, also known as an onto function, is a function where every element in the codomain is mapped to by at least one element in the domain. In other words, every element in the codomain has a preimage in the domain.

Explanation

To determine whether a function is invertible, we need to check if it is both injective and surjective. Let's analyze each option:

a) Surjective: A surjective function does not necessarily guarantee invertibility. It only ensures that every element in the codomain has a preimage in the domain. However, it does not guarantee that these preimages are unique.

c) Injective: An injective function ensures that every element in the domain is mapped to a unique element in the codomain. While being injective is a necessary condition for invertibility, it is not sufficient. A function can be injective but not surjective, meaning there are elements in the codomain that do not have a preimage in the domain.

d) Neither surjective nor injective: This option is incorrect because both injectivity and surjectivity are necessary conditions for invertibility.

Therefore, the correct answer is option b) bijective. A bijective function is both injective and surjective, guaranteeing that each element in the domain is mapped to a unique element in the codomain, and every element in the codomain has a preimage in the domain. This ensures that the function has an inverse function, making it invertible.