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January 3rd, 2018, 09:48 AM   #1
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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.
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January 3rd, 2018, 09:55 AM   #2
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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.
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January 3rd, 2018, 10:06 AM   #3
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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.
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January 3rd, 2018, 10:11 AM   #4
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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
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January 3rd, 2018, 03:29 PM   #5
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is there a reason for finding the function's inverse? i.e., is there a derivative calculation involved?
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January 3rd, 2018, 04:23 PM   #6
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No, it's just find the inverse and domain of the function and of the inverse
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January 3rd, 2018, 05:17 PM   #7
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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
File Type: jpg inverse.jpg (38.0 KB, 5 views)
Thanks from greg1313, Country Boy and ModoSK
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January 3rd, 2018, 07:43 PM   #8
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For each of those domains, the inverse function can be obtained using the Lambert W function. Take care to choose the correct branch.
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