Is the Inverse of a Function Always Well-Defined?

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Discussion Overview

The discussion revolves around the properties of the inverse of a function, specifically whether certain identities involving the function and its inverse hold true. The scope includes theoretical aspects of functions and their inverses, as well as considerations of specific examples.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\) hold true for all functions, using \(f(x) = x^2\) as an example where the inverse is not uniquely defined.
  • Another participant notes that the validity of \(f(f^{-1}(B))\) cannot be definitively stated as true or false without additional context regarding the function \(f\) and the definition of \(f^{-1}\).
  • A later reply emphasizes that the first identity holds only if \(f^{-1}\) is a single-valued function, suggesting that the uniqueness of the inverse is crucial.
  • One participant expresses confusion about the responses and requests further clarification on the points raised.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the validity of the identities involving the inverse function, with multiple competing views and uncertainties expressed throughout the discussion.

Contextual Notes

The discussion highlights the dependence on the specific properties of the function \(f\) and the definitions used for the inverse, which remain unresolved.

OhMyMarkov
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Hello everyone!

I have three questions:

(1) If $x\in R$, is it true that $f ^{-1} (f(x)) = x$?
(2) If $y\in R$, is it true that $f (f^{-1}(y)) = y$?
(3) If $B\subset R$, is it true that $f(f ^{-1} (B)$?

I think I have showed it for (3), but not sure of my proof. For (1) and (2), I considered the function $f (x) = x^2$. $f^{-1}(1)$ can be 1 and -1...

Thanks for the help!
 
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The answer may depend on what \(f\) is and on the precise definition of \(f^{-1}\). Also, in (3), \(f(f^{-1}(B))\) cannot be true or false.
 
OhMyMarkov said:
Hello everyone!

I have three questions:

(1) If $x\in R$, is it true that $f ^{-1} (f(x)) = x$?
(2) If $y\in R$, is it true that $f (f^{-1}(y)) = y$?
(3) If $B\subset R$, is it true that $f(f ^{-1} (B)$?

I think I have showed it for (3), but not sure of my proof. For (1) and (2), I considered the function $f (x) = x^2$. $f^{-1}(1)$ can be 1 and -1...

Thanks for the help!

As in Your example, the (1) supplies one and only one x if and only if $\displaystyle f^{-1} (*)$ is a single value function...

Kind regards

$\chi$ $\sigma$
 
I've returned to although I have seen this before, and I thought I was convinced.

Could you explain your answer, I don't think I understand...

Thank you.
 

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