inverse function

[woohoooo]

i need to find the inverse function of 3(x^2)(exp(-(x^3))) to solve part of my question

i first tried to split into 2 different functions, 3(x^2) and exp(-(x^3))) and find these inverses, (x/3)^(1/2) and
(-(log(x))^(1/3) and then recombined them, however this proved to be incorrect ( by checking an inverse graph image.)

i then tried to integrate the function to exp - (x^3) and then rearrange the integral, bearing in mind that y goes to yx when integrated.
I tried applying the natural log, ln in this case to simplify my rearranging however i was unable to do so.

any help or suggestions would be greatly appreciated as i have spent significant periods of time on this question to no avail.
I can easily check answers using graphs so all reasonable suggestions are welcome.
thank you very much.

 

[galactus]

This is difficult to find the inverse of. It is in terms of the Lambert W function.

\(\displaystyle \frac{1}{3}\left(-18\text{LambertW}(\pm\frac{\sqrt{3}}{6}\cdot y^{\frac{3}{2}})\right)^{\frac{1}{3}}\)

 

[BigGlenntheHeavy]

\(\displaystyle If \ your \ function \ is \ 3x^{2}e^{-x^{3}}, \ without \ restrictions, \ it \ has \ no \ inverse, \ as \ it \ isn't \ one \ to \ one.\)

[attachment=0crmvczf]rrr.jpg[/attachmentcrmvczf]

\(\displaystyle Graph \ of \ f(x) \ = \ 3x^{2}e^{-x^{3}}.\)

 

[woohoooo]

its restricted x>0

 

[BigGlenntheHeavy]

You mean x <0 as if x> 0 it is still not 1 to 1, see graph.

 

[BigGlenntheHeavy]

[attachment=0y1318ds]inv.jpg[/attachmenty1318ds]

As one can see by the above graph if f(x) < 0, we have an inverse function, however if f(x) > 0 (unless restrictions are applied) we do not.

However to find the equation of the inverse is another matter, good luck.

 

[woohoooo]

if x > 0 then it is a one - to - one function, ( or injective) as for each x there exists a unique y point in the domain.
(nb x>0 not F(x)>0)

obviously its hard, hence the need for help!

 

[DrMike]

i need to find the inverse function of 3(x^2)(exp(-(x^3))) to solve part of my question

i first rolled it with sawdust, and salted it with glue, then I condensed it with locusts and tape, keeping one principal object in view - to preserve its suppetrical shape.

Sorry for changing your quote, but are you very sure you need an inverse function for 3(x^2)(exp(-(x^3))) ? My guess is you are trying to solve an integral, using the substitution

\(\displaystyle u=3(x^2)(exp(-(x^3)))\).

Am I right?

If so, here's what I would suggest...

* You don't need the full inverse function - you only need to solve a=3x^2exp(-x^3) and b=3x^2exp(-x^3), where a and b are the bounds on the integral. This might be very easy. For example, if a=0, then x=0 or -infinity might do. Or if a=3/e, then x=1.

* Perhaps you might try a different substitution? For example, u=x^3 or u=-x^3 ?