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 (injective) if each element of the function's domain maps to a unique element in the codomain, meaning no two distinct inputs produce the same output. This property is essential for determining whether an inverse function exists.

Inverse Function

An inverse function reverses the mapping of the original function, taking outputs back to their original inputs. To construct the inverse, one typically solves the equation y = f(x) for x in terms of y, and then interchanges the variables. Importantly, only one-to-one functions have inverses that are also functions.

Domain Restriction

Domain restriction involves limiting the set of input values for a function to ensure that it possesses certain desirable properties, such as being one-to-one. For functions that are not naturally one-to-one over their entire domain, a proper restriction can allow the determination of an inverse function.

Horizontal Line Test

The horizontal line test is a visual method used to verify if a function is one-to-one. If every horizontal line intersects the graph of the function at most once, the function passes the test and is one-to-one, making it a candidate for having an inverse function.

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