If $f$ is invertible is $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x$ always True? If $f$ is invertible, is $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x$ always True?
I have doubt in the second one since $$\sin^{-1}(\sin x)=x$$ is not always true.
Any comments on this?

Yes, it is always true. It turns out that $\sin$ is not invertible. The function which is denoted by $\sin^{-1}$ (or, more generally, by $\arcsin$) is the inverse of the restriction of $\sin$ to $\left[-\frac\pi2,\frac\pi2\right]$. And so$$\left(\forall x\in\left[-\frac\pi2,\frac\pi2\right]\right):\arcsin\bigl(\sin(x)\bigr)=x$$and$$\bigl(\forall x\in[-1,1]\bigr):\sin\bigl(\arcsin(x)\bigr)=x.$$
