What characterizes a bijective function?

What characterizes a bijective function?

• A. Functions that are only surjective
• B. Functions that lack any inverse
• C. Functions that are both injective and surjective
• D. Functions that do not map all elements

Answer: (C)

A bijective function is defined as a function that is both injective and surjective. This means that every element in the domain maps to a unique element in the codomain (injective), and every element in the codomain has a pre-image in the domain (surjective). Being injective ensures that no two different elements in the domain map to the same element in the codomain. This uniqueness is crucial because it allows for the existence of an inverse function. Meanwhile, being surjective guarantees that every possible output in the codomain is covered by at least one input from the domain, making it a complete mapping. Together, these two properties—injectiveness and surjectiveness—establish a one-to-one correspondence between the domain and codomain, allowing the function to have an inverse. Thus, the correct characterization of a bijective function is that it is one which meets both of these criteria.

A bijective function is defined as a function that is both injective and surjective. This means that every element in the domain maps to a unique element in the codomain (injective), and every element in the codomain has a pre-image in the domain (surjective).

Being injective ensures that no two different elements in the domain map to the same element in the codomain. This uniqueness is crucial because it allows for the existence of an inverse function. Meanwhile, being surjective guarantees that every possible output in the codomain is covered by at least one input from the domain, making it a complete mapping.

Together, these two properties—injectiveness and surjectiveness—establish a one-to-one correspondence between the domain and codomain, allowing the function to have an inverse. Thus, the correct characterization of a bijective function is that it is one which meets both of these criteria.