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I saw a post that said that $f^{-1}(f(x))$ is always equal to $x$. Can anyone explain to me why? I tried googling but the only thing that came close to a proof is this video, but it simply solved the equation.

The equations that made me question this are $f(x) = 3x-2$ and its inverse $f^{-1}(x) = (x+2) / 3$.

Integrand
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RemiKG
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    That's the definition of inverse function. – kingW3 Sep 19 '20 at 18:36
  • See this: https://math.stackexchange.com/questions/75365/proof-ff-1x-x/75369 – User Sep 19 '20 at 18:42
  • And this one: https://math.stackexchange.com/questions/75365/proof-ff-1x-x?r=SearchResults – User Sep 19 '20 at 18:44
  • Thanks, these links really clear up my understanding. I'm currently doing exercises about composite functions and haven't yet learned that definitionally the composite function made of a function and its inverse is always equal to x so. – RemiKG Sep 19 '20 at 18:50
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    "haven't yet learned that definitionally the composite function made of a function and its inverse is always equal to x" Then what exactly did you think the inverse was? – Arthur Sep 19 '20 at 18:55
  • @RemiKG You're welcome – User Sep 19 '20 at 19:14
  • @Arthur All we learned about it was that if $y = x^2 - 2x + 3$, then its inverse would be $x = y^2 - 2y + 3$ simplified. With the links above, I can sadly only really partially understand what inverse functions mean, this one especially helped me since it makes a comparison. – RemiKG Sep 19 '20 at 19:25
  • It strongly depends whether $f^{-1}$ denotes the inverseor the preimage. – Michael Hoppe Sep 20 '20 at 12:04
  • I’m voting to close this question because there is no better answer than the one given on the day that question was asked. – Kurt G. Jan 25 '23 at 13:55

1 Answers1

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Let's say we have a function $f$ such that $y=f(x)$. If $f$ is invertible (has an inverse), this inverse $f^{-1}$ satisfies the property

$$f^{-1}(y)=x$$

We established earlier, however, that $y=f(x)$. This means that

$$f^{-1}(f(x))=x$$

where $x$ is in the domain of $f$.

This is similar to this proof that $f(f^{-1}(x))=x$.

Kman3
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