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When does a function have an inverse function?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

A mathematical function has an inverse function if and only if it is a bijection.1

A bijection is a mapping that is both one-to-one (injective) and onto (surjective). This means that every element in the function's codomain is mapped to by exactly one element in its domain. When these conditions are met, the function is considered invertible because it is possible to uniquely "undo" its effect, mapping each output back to its original input.12

If a function fails to be one-to-one—meaning multiple inputs produce the same output—or fails to be onto, it cannot have a well-defined inverse function across its entire domain.