Let x neq 0 and |x|

Question

(A) Given the function $f(x) = 1 + 2x + 4x^2 + 8x^3 + \ldots$ for $|x| < \frac{1}{2}$.This is an infinite geometric series with first term $a = 1$ and

Vedclass · Accepted Answer

$\frac{x-1}{2x}$

Vedclass · Answer

(A) Given the function $f(x) = 1 + 2x + 4x^2 + 8x^3 + \ldots$ for $|x| This is an infinite geometric series with first term $a = 1$ and common ratio $r = 2x$.The sum of an infinite geometric series is given by $S_{\infty} = \frac{a}{1-r}$.Thus, $f(x) = \frac{1}{1-2x}$.Let $y = f(x) = \frac{1}{1-2x}$.To find $f^{-1}(x)$, we solve for $x$ in terms of $y$:$y(1-2x) = 1$$y - 2xy = 1$$2xy = y - 1$$x = \frac{y-1}{2y}$.Replacing $y$ with $x$, we get $f^{-1}(x) = \frac{x-1}{2x}$.

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

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