My Math Forum inverse of e^(-x) * sqrt(x-1)

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 January 3rd, 2018, 09:48 AM #1 Newbie   Joined: Jan 2018 From: Slovensko Posts: 4 Thanks: 0 inverse of e^(-x) * sqrt(x-1) Hi, first time posting here. I need help with solving inverse of this function: function.jpg I searched the internet for some answers but couldn't find anything. I would be really thankful if you could help me. Last edited by skipjack; January 3rd, 2018 at 07:38 PM.
 January 3rd, 2018, 09:55 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Do you have any reason to think that the inverse function can be written in terms of elementary functions? My first guess would be to try to write it in terms if the Lambert W function but I don't see any obvious way.
 January 3rd, 2018, 10:06 AM #3 Newbie   Joined: Jan 2018 From: Slovensko Posts: 4 Thanks: 0 Well, to be honest I have a test in a couple of days and this is one of the questions. I have spent a lot of time trying to figure this out, and I have no clue how to solve it; that's why I posted here. The only reason to think that the inverse function can be written in terms of elementary functions is the fact that it's on the test, so I should be able to solve it... And I have no idea what a Lambert W function is. I hope you understand what I mean, English is not my native language, so maybe I didn't say everything quite right. Last edited by skipjack; January 3rd, 2018 at 07:37 PM.
 January 3rd, 2018, 10:11 AM #4 Newbie   Joined: Jan 2018 From: Slovensko Posts: 4 Thanks: 0 Well, to be honest I have it on a test in a couple of days, thats the only reason why I think it can be written in terms of elementary functions, cause I should be able to solve it, and I have no clue what is a Lambert W function
 January 3rd, 2018, 03:29 PM #5 Math Team     Joined: Jul 2011 From: Texas Posts: 2,949 Thanks: 1555 is there a reason for finding the function's inverse? i.e., is there a derivative calculation involved?
 January 3rd, 2018, 04:23 PM #6 Newbie   Joined: Jan 2018 From: Slovensko Posts: 4 Thanks: 0 No, it's just find the inverse and domain of the function and of the inverse
January 3rd, 2018, 05:17 PM   #7
Math Team

Joined: Jul 2011
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domain of $f(x)$ is $x \ge 1$

$f'(x) = e^{-x} \cdot \dfrac{3-2x}{\sqrt{x-1}}$

note ...

$f'(x) = 0$ at $x = \dfrac{3}{2}$

$f'(x) > 0$ for $1 < x < \dfrac{3}{2}$

$f'(x) < 0$ for $x > \dfrac{3}{2}$

$f(x)$ has a maximum at $x = \dfrac{3}{2}$, therefore the range of $f(x)$ is $0 \le y \le f\left(\dfrac{3}{2}\right)$

Since $f(x)$ is not 1-1, its domain must be restricted to determine an inverse.

As stated in previous posts, an explicit expression for $f^{-1}(x)$ cannot be found using elementary methods.

Graph of the two possible restricted domains is attached ...
Attached Images
 inverse.jpg (38.0 KB, 5 views)

 January 3rd, 2018, 07:43 PM #8 Global Moderator   Joined: Dec 2006 Posts: 20,757 Thanks: 2138 For each of those domains, the inverse function can be obtained using the Lambert W function. Take care to choose the correct branch. Thanks from Country Boy

 Tags inverse, sqrtx1

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