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

Which mathematician invented the inverse function notation "$$f^{-1}$$"? I can't find an answer anywhere. Why did s/he use "$$f^{-1}$$" as their notation?

• Which inverse notation? – Arthur Oct 16 '19 at 14:23
• You mean something like $f^{-1}$? Seems like a completely obvious notation, doesn't it? – lulu Oct 16 '19 at 14:23
• See Earliest Uses of Symbols of Operation : Negative integers as exponents : "Negative integers as exponents were first used with the modern notation by Isaac Newton in June 1676 in a letter to Henry Oldenburg, secretary of the Royal Society, in which he described his discovery of the general binomial theorem twelve years earlier (Cajori 1919, page 178). Before Newton, John Wallis suggested the use of negative exponents but did not actually use them (Cajori vol. 1, page 216)." – Mauro ALLEGRANZA Oct 16 '19 at 14:39
• Then we "generalize" it to functions : $f^{-1} \circ f= \text {id}$. – Mauro ALLEGRANZA Oct 16 '19 at 14:42
• It's quite woke of you to think a woman or nonbinary person may have come up with this notation. But even if that was the case, history would not record it that way. – Robert Soupe Oct 17 '19 at 14:25

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).