X is a random variable, so the associated value of the cdf that you get from that is a random variable also. If you assume that F(X) is monotone increasing over the support (which is extraordinarily common), then this problem is very straightforward: for fixed 0 < y < 1
which is the CDF for a uniform (0, 1). In F(X) is not monotone increasing (i.e. it is flat in places), then this method won't work, since F(X) won't have an inverse. It turns out that the same proof works if you define , but I wouldn't be surprised if they expected you to ignore this case.