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.

Let f : $R \to R$ be defined by $f\left( x \right) = \ln \left( {x + \sqrt {{x^2} + 1} } \right)$ , then number of solutions of $\left| {{f^{ – 1}}\left( x \right)} \right| = {e^{ – \left| x \right|}}$ is

Inverse of the function $y = 2x – 3$ is

If $f:[1,\; + \infty ) \to [2,\; + \infty )$ is given by $f(x) = x + \frac{1}{x}$ then ${f^{ – 1}}$ equals

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