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Determine whether the function has an inverse function. If...

Key Concepts

Horizontal Line Test

The horizontal line test is a visual method for determining if a function is one-to-one. If every horizontal line drawn across the function’s graph intersects it at most once, the function passes the test and is considered one-to-one. This test is particularly useful for functions that are defined by absolute value or quadratic expressions, where symmetry might produce repeated y-values.

Absolute Value Function

Absolute value functions, such as the one given in the problem, typically produce graphs that are V-shaped and symmetric about a vertical axis. This inherent symmetry often leads to the failure of the one-to-one property over the entire real line since many inputs can produce the same output, thereby complicating or preventing the existence of an inverse function without restricting the domain.

One-to-One Functions

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. This property is crucial for finding an inverse because it guarantees that the function can be reversed. If a function is not one-to-one, it will fail the horizontal line test, meaning some horizontal line would intersect the graph more than once, preventing the construction of a unique inverse.

Inverse Function

An inverse function reverses the mapping of the original function, meaning that if f(x) maps an input x to an output y, then the inverse function f?¹(y) maps y back to x. It essentially undoes the effect of the original function and exists only if the original function is one-to-one, ensuring every output is uniquely paired with one input.

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