For each function, a) determine whether it is one-to-one; b)...

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Key Concepts

One-to-One Function

A one-to-one function (or injective function) is one in which each element of the domain maps to a unique element in the codomain. No two distinct inputs produce the same output, which is a necessary property for a function to have an inverse function that is also a function.

Inverse Function

An inverse function reverses the mapping of the original function. If f is one-to-one, then there exists a function f?¹ such that for every y in the range of f, f?¹(y) returns the unique x for which f(x) = y. The process of finding an inverse typically involves swapping the roles of the dependent and independent variables and solving for the new dependent variable.

Domain and Range Restrictions

Understanding and possibly restricting the domain and range of a function is crucial in ensuring the function is one-to-one and that an inverse can be properly defined. Restrictions may be necessary to eliminate duplicate outputs, which would otherwise prevent the function from having a unique inverse.

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