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Finding an Inverse

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Polynomials that are strictly increasing or strictly decreasing have inverse functions. For example, and are strictly increasing. However, neither nor are one-to-one, and so they do not have an inverse defined for all real .

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A polynomial is one-to-one on its intervals of increase and decrease. A restriction of the polynomial is a new function, with one of those intervals as its domain, whose values agree with the values of the polynomial on that interval. Those functions are one-to-one on those intervals and have inverses. For example, the function defined for with values has the inverse function . The function defined for with values has the inverse function .

The graphs of a function and its inverse are symmetric in the line .

This Demonstration plots the graphs of each restricted function (solid curve) and its inverse (dashed curve) in matching colors.

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Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA

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