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I would like to calculate the inverse of $y = 4x^3 - 3x^4$ on the domain $x = [0,1]$.

What would be the best way to tackle this?

I'd preferably a general method, suitable for tackling other nontrivial functions.

If it is not possible to have an explicit/exact solution for this, how else would you get a drawing from a particular pdf via Matlab/Java/Python?

My idea was to use the inverse transformation method for the cdf and then use the uniform distribution with $x = F'(u)$

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If your purpose is to plot inverse functions, it suffices to swap the variables and the axis and plot the direct functions. This is much easier.

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