If $X$ and $Y$ are two non- empty sets where $f:X \to Y$ is function is defined such that $f(c) = \left\{ {f(x):x \in C} \right\}$ for $C \subseteq X$ and ${f^{ - 1}}(D) = \{ x:f(x) \in D\} $ for $D \subseteq Y$ for any $A \subseteq X$ and $B \subseteq Y,$ then

Let $S=\{a, b, c\}$ and $T=\{1,2,3\} .$ Find $F^{-1}$ of the following functions $F$ from $S$ to $T$. if it exists.  $F =\{( a , 3),\,( b , 2),\,( c , 1)\}$

Consider $f: R _{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4 .$ Show that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}(y)=\sqrt{y-4},$ where $R ^{+}$ is the set of all non-negative real numbers.

State with reason whether following functions have inverse $g :\{5,6,7,8\} \rightarrow\{1,2,3,4\}$ with $g=\{(5,4),(6,3),(7,4),(8,2)\}$

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

Which of the following functions is inverse of itself