Let $f : R \rightarrow R$ be defined by $f(x) = x^3 - 3x^2 + 3x - 2$. Then $f^{-1}(x)$ is given by:

If $f(x) = x + \tan x$ and $g(x)$ is the inverse of $f(x)$,then $g'(x)$ is equal to

Let $f: R - \{-\frac{4}{3}\} \rightarrow R$ be a function defined as $f(x) = \frac{4x}{3x+4}$. The inverse of $f$ is the map $g: \text{Range } f \rightarrow R - \{-\frac{4}{3}\}$ given by

Let $f: R-\{2\} \rightarrow R-\{1\}$ defined by $f(x)=\frac{x-3}{x-2}$ and $g: R \rightarrow R$ defined by $g(x)=3x-2$. Then,the sum of all values of $x$ for which $f^{-1}(x)+g^{-1}(x)=\frac{19}{6}$ is

Let $x \neq 0$ and $|x| < \frac{1}{2}$. If $f(x) = 1 + 2x + 4x^2 + 8x^3 + \ldots$, then $f^{-1}(x) =$

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