For $f(x)=\frac{4}{x+4}$ and $g(x)=\frac{3}{x}$, find
a. $(f \circ g)(x)$;
b. the domain of $f \circ g$
a. $(f \circ g)(x)=\frac{4 x}{3+4 x}$
(Simplify your answer.)
b. What is the domain of $f \circ g$ ?
The domain is
(Simplify your answer. Type your answer in interval notation. the expression.)

Step 1 ：Given the functions $f(x)=\frac{4}{x+4}$ and $g(x)=\frac{3}{x}$, we are asked to find the composition of $f$ and $g$, denoted as $(f \circ g)(x)$, and the domain of this composition.

Step 2 ：To find the composition of two functions, we substitute the second function into the first. In this case, we substitute $g(x)$ into $f(x)$, which gives us $f(g(x)) = f\left(\frac{3}{x}\right) = \frac{4}{\frac{3}{x} + 4}$.

Step 3 ：Simplifying this expression, we get $(f \circ g)(x) = \frac{4x}{4x + 3}$.

Step 4 ：The domain of a function is the set of all real numbers for which the function is defined. In this case, the function will be undefined when the denominator is zero. So, we need to find the values of $x$ for which $4x + 3 = 0$.

Step 5 ：Solving the equation $4x + 3 = 0$, we find that $x = -\frac{3}{4}$.

Step 6 ：Therefore, the function is undefined when $x = -\frac{3}{4}$. Hence, the domain of the function is all real numbers except $-\frac{3}{4}$.

Step 7 ：\(\boxed{(f \circ g)(x)=\frac{4x}{4x + 3}}\)

Step 8 ：\(\boxed{\text{The domain of } f \circ g \text{ is } (-\infty, -\frac{3}{4}) \cup (-\frac{3}{4}, \infty)}\)