For the functions $f(x)=\sqrt{x-4}$ and $g(x)=5 x$ find the following.
(a) $(f \circ g)(x)$ and its domain
(b) $(g \circ f)(x)$ and its domain
$(a)(f \circ g)(x)=$
(Simplify your answer.)
The domain is
(Type your answer in interval notation.)
(b) $(g \circ f)(x)=$
(Simplify your answer.)
The domain is
(Type your answer in interval notation.)

Step 1 ：First, we find the composition of the functions $(f \circ g)(x)$ and $(g \circ f)(x)$.

Step 2 ：For $(f \circ g)(x)$, we substitute $g(x)$ into $f(x)$, so we get $f(g(x)) = f(5x) = \sqrt{5x - 4}$.

Step 3 ：The domain of $(f \circ g)(x)$ is the set of all $x$ such that $5x - 4 \geq 0$. Solving this inequality gives $x \geq \frac{4}{5}$. So, the domain of $(f \circ g)(x)$ is $[\frac{4}{5}, \infty)$.

Step 4 ：For $(g \circ f)(x)$, we substitute $f(x)$ into $g(x)$, so we get $g(f(x)) = g(\sqrt{x - 4}) = 5\sqrt{x - 4}$.

Step 5 ：The domain of $(g \circ f)(x)$ is the set of all $x$ such that $x - 4 \geq 0$. Solving this inequality gives $x \geq 4$. So, the domain of $(g \circ f)(x)$ is $[4, \infty)$.

Step 6 ：Finally, we have $(f \circ g)(x) = \sqrt{5x - 4}$ with domain $[\frac{4}{5}, \infty)$ and $(g \circ f)(x) = 5\sqrt{x - 4}$ with domain $[4, \infty)$.