Let $S =\{1,2,3\} .$ Determine whether the functions $f: S \rightarrow S$ defined as below have inverses. Find $f^{-1}$, if it exists. $f=\{(1,1),\,(2,2),\,(3,3)\}$

The inverse function of $f(\mathrm{x})=\frac{8^{2 \mathrm{x}}-8^{-2 \mathrm{x}}}{8^{2 \mathrm{x}}+8^{-2 \mathrm{x}}}, \mathrm{x} \in(-1,1),$ is

Let $f: X \rightarrow Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$, i.e., $\left(f^{-1}\right)^{-1}=f$.

Let $Y =\left\{n^{2}: n \in N \right\} \subset N .$ Consider $f: N \rightarrow Y$ as $f(n)=n^{2}$  Show that $f$ is invertible. Find the inverse of $f$

Let $f:\left[ {4,\infty } \right) \to \left[ {1,\infty } \right)$ be a function defined by $f\left( x \right) = {5^{x\left( {x – 4} \right)}}$  then $f^{-1}(x)$  is