I have an exercise in a basic simulation course asking me to use the inverse function method to describe an algorithm to generate a sample of a random variable whose distribution function is given.
The method is sampling a uniform variable $U\sim U(0,1)$ and then return $F^{-1}(u)$ where $u$ is the sampled value.
I'm having a technical difficulty finding the inverse of the distribution function, which is defined differently on different intervals and on $[0,1]$ is defined by $$\frac{1}{2}x^{3}+\frac{1}{2}(1-e^{-x})$$
I tried to solve $$\frac{1}{2}x^{3}+\frac{1}{2}(1-e^{-x})$$
and got stuck at $$x^{3}-e^{-x}=2y-1$$
I would appreciate help continuing further