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-   -   inverse of e^(-x) * sqrt(x-1) (http://mymathforum.com/calculus/343171-inverse-e-x-sqrt-x-1-a.html)

ModoSK January 3rd, 2018 09:48 AM

inverse of e^(-x) * sqrt(x-1)
 
1 Attachment(s)
Hi, first time posting here. :)
I need help with solving inverse of this function:

Attachment 9364

I searched the internet for some answers but couldn't find anything. I would be really thankful if you could help me.

Country Boy January 3rd, 2018 09:55 AM

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.

ModoSK January 3rd, 2018 10:06 AM

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. :D

I hope you understand what I mean, English is not my native language, so maybe I didn't say everything quite right. :D

ModoSK January 3rd, 2018 10:11 AM

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 :D

skeeter January 3rd, 2018 03:29 PM

is there a reason for finding the function's inverse? i.e., is there a derivative calculation involved?

ModoSK January 3rd, 2018 04:23 PM

No, it's just find the inverse and domain of the function and of the inverse

skeeter January 3rd, 2018 05:17 PM

1 Attachment(s)
http://mymathforum.com/attachments/c...1-function.jpg

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 ...

skipjack January 3rd, 2018 07:43 PM

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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