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As I was studying this function, $f(x)=\sqrt[3]{1-x^3}$, I checked that the function is one-to-one, and so is invertible.

Then: $$y=\sqrt[3]{1-x^3}$$

$$y^3=1-x^3$$

$$y^3-1=-x^3$$

$$1-y^3=x^3$$

$$x=\sqrt[3]{1-y^3}$$ $$f^{-1}(y)=\sqrt[3]{1-y^3}$$

What kind of functions are inverse of itselfs? Thanks.

amWhy
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1 Answers1

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A function that is its own inverse is called an involution.

ADDED: As @Mufasa puts nicely in the comment below: A functions is an involution if and only if it is symmetric about the line $y = x$, with the simplest involution given by $y = x$.

Just to nitpick: in your representation of $f^{-1}$, you need to swap $y$ and $x$ in the last step and express $f^{-1}$ in terms of $x$): $$f(x) = f^{-1}(x) = \sqrt[\large 3]{1 - x^3}$$

so that $$f(f(x)) = f(f^{-1}(x)) = x.$$

amWhy
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    I believe all involutions are symmetric about the line y=x (see here) – Mufasa Nov 23 '13 at 14:15
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    I learned something new from this answer, but with all respect, the OP asks "What kind of functions are inverse of itselfs?", this answer just gives a name to those functions, no further insight. – hhsaffar Nov 23 '13 at 14:21
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    -1 for the reason explained in hhsaffar's comment. – user104254 Nov 23 '13 at 14:40
  • @user104254 And exactly 16 minutes *before* your comment, I had already elaborated on the initial answer. Lighten up: There really isn't much more to be said, as an exhaustive list of all such functions is not possible! (There are uncountably many) ;-) – amWhy Nov 23 '13 at 15:08
  • @amWhy: Thank you for your reply. From the links to the question cited in my comment to the OP, you may see this question has been asked several times in this website. At least when the involution $f:\Bbb R\to\Bbb R$ is continuous, we can easily get an exhaustive list: it is either the identity or conjugate to $x\mapsto -x$. Unfortunately it is too late for me to cancel my downvote now. Would you mind editing your answer so that I can cancel it? – user104254 Nov 23 '13 at 15:20
  • OK. Downvote canceled, but to be honest, I have to admit that your answer still looks more like a comment than an answer to me. :) – user104254 Nov 23 '13 at 15:31
  • @amWhy: Nice answer badge on the way +1 – Amzoti Nov 25 '13 at 00:38