Determine whether each function is a one-to-one function....

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Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is one in which each element of the function's domain maps to a unique element in its range. No two different inputs produce the same output, ensuring that the function's inverse, if it exists, is also a function.

Horizontal Line Test

The horizontal line test is a graphical method used to determine whether a function is one-to-one. If every horizontal line crosses the graph of the function at most once, the function is one-to-one. This test is particularly useful when evaluating functions visually.

Domain Restriction

Restricting the domain of a function can change its behavior with respect to being one-to-one. By limiting the range of inputs to an interval where the function does not produce repeated output values, a function that might otherwise be non-injective can be made one-to-one.

Inverse Functions

A function has an inverse if and only if it is one-to-one, because only then does every output correspond to exactly one input. This connection between one-to-one functions and inverse functions underpins much of the study of function composition and algebraic manipulations.

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