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I know that $f^{-1}$ denotes the inverse of the function $f$. Like many others, I find this notation bizarre and ambiguous.

But my question is: where does this notation come from? Who used this notation first? In what context and what was their justification? Is there a sense in which $1/f$ really is the inverse of $f$? Did the inventor have some notion of $f^{-2}$, and if so, what would it be? (The "second inverse of $f$"?) Where there other competing notations that lost out for some reason?

My hope is that, if I can understand the notation historically, it might seem less mad. Maybe the notation even hides something profound that I don't understand.

jameshfisher
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  • Quite often the origin is a "generalization" from the number symbolism... We have to start comparing $\dfrac a a =1$ with $a a^{-1}=1$ and we get $a^{-1} = \dfrac 1 a$ – Mauro ALLEGRANZA Jul 13 '21 at 11:47
  • See also https://math.stackexchange.com/questions/109942/notation-for-image-and-preimage – lhf Jul 13 '21 at 12:03
  • @MauroALLEGRANZA the problem I have with that theory is - why would we choose the multiplicative inverse notation? Couldn't we instead say $-f$ is the inverse of $f$, by analogy with additive inverse? Berci's theory that it derives from function composition seems more convincing ... – jameshfisher Jul 13 '21 at 13:24

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Well, function composition is the primary operation among generic functions, say $X\to X$ for a set $X$, which is actually a semigroup operation, i.e. it is associative: $f\circ (g\circ h)=(f\circ g)\circ h$, moreover the identity function on $X$ is the identity element.

The inverse of a function $X\to X$ is its inverse in this semigroup.

When there is no multiplication in context, one can simply write $fg$ for $f\circ g$, $f^2$ for $f\circ f$, and then $f^{-2}$ is just $f^{-1}\circ f^{-1}$.

Berci
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  • a-HA, that must be it! I hadn't seen this $f^2$ notation before! I now see a history of it here. And $f^{-1}$ being the inverse makes perfect sense with this notation, and gives meaning to $f^{-2}$ etc like you say. Great! What's strange is that this notation was never taught to me - the notation $f^{-1}$ just came out of nowhere ... – jameshfisher Jul 13 '21 at 13:14
  • It could even give meaning to other values e.g. $f^{1/2}$ ... "a/the function that when done twice is the same as $f$" ... etc. This makes so much more sense than isolated $f^{-1}$. – jameshfisher Jul 13 '21 at 13:21
  • Wikipedia suggests the more explicit notation $f^{\circ n}$ which parameterizes by the underlying operation ($\circ$, function composition). So I suppose ordinary $x^{n}$ is shorthand for $x^{\times n}$ using this same notation ... – jameshfisher Jul 13 '21 at 13:26