For the functions $f(x)=\frac{2}{x+3}$ and $g(x)=\frac{13}{x+3}$, find the composition $f \circ g$ and simplify your answer as much as possible. Write the domain using interval notation.

Step 1 ：Given the functions \(f(x)=\frac{2}{x+3}\) and \(g(x)=\frac{13}{x+3}\), 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, \(f \circ g\), is defined as \((f \circ g)(x) = f(g(x))\). So, to find \(f \circ g\), we need to substitute \(g(x)\) into \(f(x)\).

Step 3 ：Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = \frac{2}{3 + \frac{13}{x + 3}}\).

Step 4 ：Simplifying the above expression, we get \(f(g(x)) = \frac{2*(x + 3)}{3*x + 22}\).

Step 5 ：The domain of a function is the set of all possible input values (x-values) which will produce a valid output. To find the domain of \(f \circ g\), we need to consider the domains of both \(f(x)\) and \(g(x)\), and also consider any additional restrictions that the composition might introduce.

Step 6 ：The domain is all real numbers except for those that make the denominator equal to zero. So, we need to solve the equation \(3*x + 22 = 0\) to find the values that are not in the domain.

Step 7 ：Solving the equation \(3*x + 22 = 0\), we get \(x = -\frac{22}{3}\). Therefore, the domain of \(f \circ g\) is all real numbers except \(-\frac{22}{3}\). In interval notation, this is \((-\infty, -\frac{22}{3}) \cup (-\frac{22}{3}, \infty)\).

Step 8 ：\(\boxed{\text{Final Answer: The composition } f \circ g \text{ is } \frac{2*(x + 3)}{3*x + 22} \text{ and its domain is } (-\infty, -\frac{22}{3}) \cup (-\frac{22}{3}, \infty)}\)