MHB Is the Inverse of a Function Always Well-Defined?

[OhMyMarkov]

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!

 

[Evgeny.Makarov]

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.

 

[chisigma]

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

$\chi$ $\sigma$

 

[OhMyMarkov]

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.