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

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Key Concepts

One-to-One Function

A function is one-to-one if and only if each element of the function's domain maps to a unique element in its range. This means that if f(x1) = f(x2), then x1 must equal x2. This property is crucial for ensuring that every output is tied to only one input, which is a necessary condition for the existence of an inverse function.

Inverse Function

An inverse function essentially reverses the process of the original function. For a function that is one-to-one, the inverse function f?¹ satisfies the property that f?¹(f(x)) = x for every x in the domain of f. Determining the inverse typically involves swapping the roles of the independent and dependent variables and then solving for the new dependent variable.

Domain and Range

Understanding the domain and range of a function is essential when discussing one-to-one functions and their inverses. The domain is the set of all permissible inputs for a function, while the range is the set of those inputs' corresponding outputs. When finding the inverse, the domain and range are essentially interchanged, meaning the original function's range becomes the domain of the inverse, and vice versa.

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