Inverse of $f(x) = x + e^{2x}$

Asked Oct 23 '13 at 02:38

Active Oct 23 '13 at 04:03

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How do you find the inverse of $f(x) = x + e^{2x}$ ?

I started by trying to find the inverse by replacing $f(x)$ with $y$, switching $x$ and $y$, and solving for $y$.

$$y = x + e^{2x} \implies x = y + e^{2y}$$

I took the natural log of both sides and got

$$\ln(x) = \ln(y) + 2y$$

and now I'm stuck.

edited Oct 23 '13 at 04:03

Gyu Eun Lee

asked Oct 23 '13 at 02:38

user102658

4

First, $\ln (y+e^{2y})\neq \ln y + \ln (e^{2y})=\ln y + 2y$, so you have an error there. Second, there is no way to get the inverse of this function using familiar functions. – vadim123 Oct 23 '13 at 02:42
1

Thank you! That actually helps a lot – user102658 Oct 23 '13 at 02:48
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Check the Lamber W function to solve the equation. – Mhenni Benghorbal Oct 23 '13 at 03:10
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http://www.wolframalpha.com/input/?i=inverse+of+x+%2B+e^%282x%29 – Benjamin Dickman Oct 23 '13 at 03:24
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As Mhenni Benghorbal suggested, the inverse involves a Lambert W function and the result is : x = y - W[2 e^(2y)]/2. – Claude Leibovici Oct 23 '13 at 09:15

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