Let $f(x) = 3x^3 + 4e^x$ and $g(x) = f^{-1}(x)$. The value of $g'(4)$ is equal to...

Let $f: [4, \infty) \to [1, \infty)$ be a function defined by $f(x) = 5^{x(x - 4)}$. Then $f^{-1}(x)$ is:

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

Let $g$ be the inverse function of $f$ and $f'(x) = \frac{x^{10}}{1 + x^2}$. If $g(2) = a$,then $g'(2)$ is equal to:

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

Let $f(x) = \sin x + \cos x$ and $g(x) = x^2 - 1$. Then $g(f(x))$ is invertible for $x \in $