What is a direct outcome of having a function that is both injective and surjective?

What is a direct outcome of having a function that is both injective and surjective?

• A. The function has an inverse that is also a function
• B. The function is discontinuous
• C. The function does not change outputs
• D. The function cannot be graphed

Answer: (A)

A function that is both injective (one-to-one) and surjective (onto) is referred to as a bijection. One of the key properties of bijective functions is that they have an inverse, and this inverse is also a function. This is because each output is associated with exactly one input, ensuring that the reverse mapping is well-defined and unique for every pair. For a function to have an inverse that is also a function, it must meet these criteria of injectivity and surjectivity. If a function is injective, no two different inputs will map to the same output; if it is surjective, every possible output in the codomain is produced by some input from the domain. Together, these properties ensure that the inverse operation will also produce a valid function without any ambiguities. In contrast, the other choices do not accurately describe the implications of a function being both injective and surjective. For instance, the notion of discontinuity, lack of output variance, or graphing issues is not inherently related to a bijective function. Hence, having a function that is both injective and surjective directly leads to the conclusion that it possesses an inverse function as well.

A function that is both injective (one-to-one) and surjective (onto) is referred to as a bijection. One of the key properties of bijective functions is that they have an inverse, and this inverse is also a function. This is because each output is associated with exactly one input, ensuring that the reverse mapping is well-defined and unique for every pair.

For a function to have an inverse that is also a function, it must meet these criteria of injectivity and surjectivity. If a function is injective, no two different inputs will map to the same output; if it is surjective, every possible output in the codomain is produced by some input from the domain. Together, these properties ensure that the inverse operation will also produce a valid function without any ambiguities.

In contrast, the other choices do not accurately describe the implications of a function being both injective and surjective. For instance, the notion of discontinuity, lack of output variance, or graphing issues is not inherently related to a bijective function. Hence, having a function that is both injective and surjective directly leads to the conclusion that it possesses an inverse function as well.