For the functions $f(x)=\frac{4}{x-3}$ and $g(x)=\frac{13}{x}$, find the composition $f \circ g$ and simplify your answer as much as possible. Write the domain using interval notation.
\[
(f \circ g)(x)=\square
\]
Domain of $f \circ g: \square$

Step 1 ：Given the functions \(f(x)=\frac{4}{x-3}\) and \(g(x)=\frac{13}{x}\), we are asked to find the composition \(f \circ g\) and simplify the answer as much as possible.

Step 2 ：The composition of two functions, \(f(g(x))\), means that we substitute \(g(x)\) into \(f(x)\). So, we substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\).

Step 3 ：Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = f(\frac{13}{x}) = \frac{4}{\frac{13}{x}-3}\).

Step 4 ：Simplifying the above expression, we get \(f(g(x)) = -\frac{4x}{3x - 13}\).

Step 5 ：The domain of a function is the set of all possible input values (x-values) which will output real numbers. To find the domain of the composition of two functions, we need to consider the domains of both the original functions and the composition.

Step 6 ：The domain of a rational function is all real numbers except those that make the denominator equal to zero. So, we need to find the values of \(x\) that make \(3x - 13 = 0\).

Step 7 ：Solving the equation \(3x - 13 = 0\), we find that \(x = \frac{13}{3}\).

Step 8 ：Therefore, the domain of \(f(g(x))\) is all real numbers except \(\frac{13}{3}\).

Step 9 ：Also, we need to consider the domain of \(g(x)\), which is all real numbers except \(0\).

Step 10 ：So, the domain of \(f \circ g\) is all real numbers except \(0\) and \(\frac{13}{3}\).

Step 11 ：Final Answer: \(\boxed{(f \circ g)(x) = -\frac{4x}{3x - 13}}\)

Step 12 ：Domain of \(f \circ g\): \(\boxed{(-\infty, 0) \cup (0, \frac{13}{3}) \cup (\frac{13}{3}, \infty)}\)