Where did the inverse function notation "$f^{-1}$" come from?

math-history notation

Solution 1:

From Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions:

According to Cajori (vol. 2, page 176) the inverse trigonometric function notation utilizing the exponent -1 was introduced by John Frederick William Herschel in 1813 in the Philosophical Transactions of London. A full-page footnote explained his choice of notation for the inverse trigonometric functions, such as $\cos.^{-1} e$, which he used in the body of the article (Cajori vol. 2, page 176).

However, according to Differential and Integral Calculus (1908) by Daniel A. Murray, "this notation was explained in England first by J. F. W. Herschell in 1813, and at an earlier date in Germany by an analyst named Burmann. See Herschell, A Collection of Examples of the Application of the Calculus of Finite Differences (Cambridge, 1820), page 5, note."

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