Random generation for a particular distribution

I have two probability density functions f1 and f2 that I have combined in a function f using credibility theory. Now I want to generate random points using f, how can I do that?

Moreover, suppose I do not have the density función, but I have the cumulative distribution function or its inverse, is it also possible to generate random numbers?

I know that if the distribution is standard (normal, gamma, weibull, etc.) I can use rnorm, etc., and the problem is solved. But I do not know how to do it if the density function is not one of those.