Which property must a function have to possess a valid inverse on its entire domain?

Which property must a function have to possess a valid inverse on its entire domain?

• A. It must be one-to-one
• B. It must be even
• C. It must be periodic
• D. It must be continuous

Answer: (A)

A function must be one-to-one (injective) to have a valid inverse on its entire domain. The key idea is that every output y should come from a unique input x. If two different inputs produce the same output, there would be more than one x for a given y, so the inverse would not be a function. When the function is one-to-one, you can define the inverse by mapping each y in the range back to the unique x that produced it. Other properties like being even, periodic, or merely continuous don’t guarantee this uniqueness. For example, cosine is continuous, even, and periodic, but on its entire domain it’s not one-to-one, so it doesn’t have a proper inverse there.

A function must be one-to-one (injective) to have a valid inverse on its entire domain. The key idea is that every output y should come from a unique input x. If two different inputs produce the same output, there would be more than one x for a given y, so the inverse would not be a function. When the function is one-to-one, you can define the inverse by mapping each y in the range back to the unique x that produced it.

Other properties like being even, periodic, or merely continuous don’t guarantee this uniqueness. For example, cosine is continuous, even, and periodic, but on its entire domain it’s not one-to-one, so it doesn’t have a proper inverse there.