Let $f: A \rightarrow B$ and $g: B \rightarrow A$ be defined as $f(x)=x^2 \forall x \in A$ and $g(x)=x^{1/2} \forall x \in B$. $f(x)$ and $g(x)$ are inverse functions to each other when

If $g$ is the inverse of $f$ and $f'(x) = \frac{1}{1 + x^5}$,then $g'(x) =$

The inverse of $y = 5^{\log x}$ is

Let $f(x) = (x + 1)^2 - 1$ for $x \ge -1$. Then the set $S = \{ x : f(x) = f^{-1}(x) \}$ is

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ and $g: R \rightarrow (0, 4)$ be functions defined by $f(x) = \log_e(x^2 + 2x + 4)$ and $g(x) = \frac{4}{1 + e^{-2x}}$. Define the composite function $h(x) = (f \circ g^{-1})(x)$,where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $h(x)$ at $x = 2$ is:

Which of the following functions cannot have their inverse defined? (where $[.] \to$ greatest integer function)