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

Let $f: N \rightarrow Y$ be a function defined as $f(x) = 4x + 3$,where $Y = \{y \in N : y = 4x + 3\}$ for some $\{x \in N\}$. Show that $f$ is invertible. Find the inverse.

If $f: R \rightarrow R$ is given by $f(x)=7x+8$ and $f^{-1}(12)=\frac{k}{7}$,then the value of $k$ is

Let $f(x) > 0$ for all $x$ and $f^{\prime}(x)$ exists for all $x$. If $f$ is the inverse function of $h$ and $h^{\prime}(x) = \frac{1}{1 + \log x}$, then $f^{\prime}(x)$ will be

Let $f(x) = (x - 1)^2 + 1$ for $x \ge 1$.
Statement-$1$: $S = \{x : f(x) = f^{-1}(x)\} = \{1, 2\}$.
Statement-$2$: $f$ is a bijection and $f^{-1}(x) = 1 + \sqrt{x - 1}$ for $x \ge 1$.

If $f:[1, \infty) \rightarrow [1, \infty)$ is defined by $f(x) = \frac{1+\sqrt{1+4 \log_2 x}}{2}$, then $f^{-1}(3) =$