Question 1 : What's the difference between $(f^{-1}of)(x)$ and$(fof^{-1})(y)$ ? Also can we say $(f^{-1}of)(x) = (fof^{-1})(x) = x$ ?
Question 2 : Why these notations are different ? $\arctan(x)$ , $arc(\frac{sin(x)}{cos(x)})$ and $\frac{\arcsin(x)}{\arccos(x)}$
Question 3 : How we can find intersection points of $f(x)$ and $f^{-1}(x)$ or how we can solve $f(x) = f^{-1}(x)$ ?
My try : I really don't know any trustable source for finding the answers.
On
In general we do not have $(f^{-1}of)(x) = (fof^{-1})(x) = x$
Example: Let $A=\{1,2\}$, $B=\{3,4\}$ and let $f :A \to B$ be defined by
$f(1)=3$, $f(2)=4$.
Then $f$ is bijective and $f^{-1}:B \to A$ is given by
$f^{-1}(3)=1$ and $f^{-1}(4)=2$.
We have $(f^{-1}of)(a) = a$ for all $a \in A$ and $(fof^{-1})(b) = b$ for all $b \in B$ .
Observe that $A \cap B= \emptyset$.
Case $1$: For functions $f:X → Y$ and $f^{−1}:Y → X$, $${\displaystyle f^{-1}\circ f=\mathrm {id} _{X}} \text { and } {\displaystyle f\circ f^{-1}=\mathrm {id} _{Y}.}$$ where $\mathrm {id}_x $ is the identity function on the set $X$. So there is no difference.
Case $2$: It's true that $\frac {\sin x}{\cos x}=\tan x $, but the functions $\arcsin x, \arccos x$ and $\arctan x$ are the inverse functions of the former. If you think about it, it actually makes more sense that if some relation is true for a set of functions, it will not be true for their inverse functions.
Case $3$: Has been discussed many times on the site, for example see here.
Hope it helps.