$A_f \otimes_A A_g = A_{fg} $

Question

A is a commutative ring. $f,g\in A$. Then I want to show $A_f \otimes_A A_g = A_{fg}$. I tried proving it by constructing a bilinear form from $A_f \times A_g$ to $A_{fg}$. But I could not figure out how the bilinear form would look like. Is there any other way to do this problem?