Prove that $f$ is injective if and only if $f^{-1}(f(x)) = x \; \; \forall x \in A$ [closed]

I want to prove that a function $f$ is injective if and only if $f^{-1}(f(x)) = x \; \; \forall x \in A.$ I don't know how to solve it. Can anyone please help me prove it.

Solution 1:

First Recall the definition of an Injective function (I prefer the term one-one function):

A function $f:A \rightarrow \textbf{R} $ is said to be one-one if $ \forall x,y \in A, \; f(x)=f(y) \Rightarrow x=y $

Now, assume that $f^{-1}(f(x)) =x \; \; \; \forall x \in A $. Now consider \begin{eqnarray*} f(x)&=&f(y) \\ \Rightarrow f^{-1}(f(x)) &=& f^{-1}(f(y)) \\ \Rightarrow x&=&y \end{eqnarray*}

Can you prove the other way now?