If ${e^x} = y + \sqrt {1 + {y^2}} $,then $y =$

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

If a function $f:(-1,1) \rightarrow B(\subseteq R)$ is defined as $f(x)=x+x^2+x^3+\ldots \infty$, then in order to have the inverse function of $f$, $B$ is equal to

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

Let $f: R - \{3\} \rightarrow R - \{1\}$ be defined by $f(x) = \frac{x-2}{x-3}$. Let $g: R \rightarrow R$ be given as $g(x) = 2x - 3$. Then,the sum of all the values of $x$ for which $f^{-1}(x) + g^{-1}(x) = \frac{13}{2}$ is equal to ...... .

Consider $f: R \rightarrow R$ given by $f(x)=4x+3$. Show that $f$ is invertible. Find the inverse of $f$.