Assume that $B \sim U[0,\bar{B}]$, and $A \sim U[0,1]$, which are independent as well. Now, I want to calculate the density of $Z = B - \frac{kB}{A}$, where $k>0$. Then, I define new random variables $(Y,Z)$ as $Y = B$ and $Z$ as above. So, $A(Y,Z) = \frac{Yk}{Y-Z}$ and $B(Y,Z) = Y$. The range of $Y$ is obvious, but what would be the $Z$'s one?
If I can find the range, I would like to calculate density of Z. Jacobian becomes $-\frac{Y-k}{(Y-Z)^2}$ It would be something like that $$f(Z) = \int |J|\frac{1}{\bar{B}} 1 dy.$$