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.

 

[blue_raver22]

Hello! I would like to clarify some points regarding the inverse of a function.

Firstly, the inverse of a function is defined as a function that "undoes" the original function. In other words, it takes the output of the original function and gives back the input. So, for (1) and (2), the statements are true if and only if $f$ is a one-to-one function, meaning that each input has a unique output.

Secondly, the statement in (3) is not always true. It depends on the specific function $f$ and the set $B$. If $f$ is not one-to-one, then $f^{-1}(B)$ may not be well-defined, and therefore, $f(f^{-1}(B))$ may not equal $B$.

For the example given, $f(x)=x^2$ is not a one-to-one function, so the statements in (1) and (2) are not always true. However, for (3), if we take $B=\{1\}$, then $f^{-1}(B)=\{1,-1\}$ and $f(f^{-1}(B))=\{1\}$. Therefore, it is not true that $f(f^{-1}(B))=B$ for all sets $B$.

I hope this helps clarify the concept of the inverse of a function. It is important to consider the properties of the specific function in order to determine the truth of these statements.