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

Let $f: N \rightarrow R$ 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 $g(x)$ is the inverse of the function $f(x)$ and $f^{\prime}(x) = \frac{1}{h(x)}$,then $g^{\prime}(x) = $

Let $f: R \rightarrow R$ be defined by $f(x) = 2x + 6$,which is a bijective mapping. Then $f^{-1}(x)$ is given by:

Let $f: W \rightarrow W$ be defined as $f(n) = n-1$ if $n$ is odd and $f(n) = n+1$ if $n$ is even. Show that $f$ is invertible. Find the inverse of $f$. Here,$W$ is the set of all whole numbers.

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 ...... .