Let $f$ be a function of 2 variables, $x$ and $a$. I do not specify a regularity class for $f$ as I don't want to restrict the question. We can assume that $f$ is as regular as we need (even analytic). Let $f(x,a)$ have a single root with respect to $x$ for every values of $a$ and let $r(a)$ be this root, i.e. $f(r(a),a)=0$.
Now let's assume that $a$ is actually a random variable with known distribution. What can we say about the distribution of $r$?
I expect that in the general case we cannot really get a closed form of the distribution. Can we get the first few moments though? Is there a standard way to do that?