Verify has an inverse, and find .
So I take the derivative and get .
It's negative for all x, so it's always decreasing, so it's 1-to-1 and has an inverse.
Now, how do I find ? Is this the same as ?
Sorry, I just wanted to give a 'general' solution to calculate the inverse function, but indeed in this case that would be really hard because of the third degree, if you have a first degree or second degree function it's less complicated.
In this case it's not very hard. We have the equation i.e. . Then, which implies the equation has only one real root. So, using the well known formula for the roots:
We can inmediately verify that the product of the two real cubic roots of the above addends is . So, we can express
where we choose for each addend the real cubic root.