If $g(x)$ is the inverse of the function $f(x)$ and $f^{\prime}(x) = \frac{1}{h(x)}$,then $g^{\prime}(x) = $

The inverse of the function $y = \frac{10^x - 10^{-x}}{10^x + 10^{-x}} + 1$ is $x =$

If $f(x) = 2 + |\sin^{-1} x|$ and $A = \{x \in R : f^{-1}(x) \text{ exists} \}$,then $A = $

Let $f: N \rightarrow R$ be a function defined as $f(x)=4x^{2}+12x+15$. Show that $f: N \rightarrow S$,where $S$ is the range of $f$,is invertible. Find the inverse of $f$.

If $f:[1, \infty) \rightarrow[5, \infty)$ is given by $f(x)=3x+\frac{2}{x}$, then $f^{-1}(x)=$

Consider the function $y = \log_{a}(x + \sqrt{x^{2} + 1})$ where $a > 0$ and $a \neq 1$. The inverse of the function: