For the real-valued functions $g(x)=\sqrt{3 x+15}$ and $h(x)=x-1$, find the composition $g \circ h$ and specify its domain using interval notation. \[ (g \circ h)(x)= \] Domain of $g \circ h$ :

Step 1 ：Define the functions \(g(x) = \sqrt{3x + 15}\) and \(h(x) = x - 1\).

Step 2 ：The composition of two functions, \(g \circ h\), is defined as \(g(h(x))\). So, we need to substitute \(h(x)\) into \(g(x)\).

Step 3 ：The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For the function \(g(x) = \sqrt{3x + 15}\), the domain is all x such that \(3x + 15 \geq 0\), because the square root of a negative number is not a real number. For the function \(h(x) = x - 1\), the domain is all real numbers, because any real number can be substituted into the function.

Step 4 ：When we compose the functions, the domain of \(g \circ h\) will be all x such that \(h(x)\) is in the domain of \(g\). So, we need to find all x such that \(3(h(x)) + 15 \geq 0\).

Step 5 ：The composition of the functions \(g\) and \(h\) is \(g(h(x)) = \sqrt{3(x-1) + 15} = \sqrt{3x + 12}\). The domain of this function is all x such that \(3(x-1) + 15 \geq 0\), which simplifies to \(x \geq -4\). Therefore, the domain of \(g \circ h\) is \([-4, \infty)\).

Step 6 ：Final Answer: \(\boxed{(g \circ h)(x) = \sqrt{3x + 12}}\)

Step 7 ：Domain of \(g \circ h\) : \(\boxed{[-4, \infty)}\)