For $f(x)=x^{2}+6$ and $g(x)=\sqrt{x-3}$, find the following composite functions and state the domain of each.
(a) $f \circ g$
(b) $g \circ f$
(c) $f \circ f$
(d) $g \circ g$
(a) $(f \circ g)(x)=\square($ Simplify your answer. $)$
Answer
Final Answer: \(\boxed{f \circ g(x) = x + 3}\) for \(x \geq 3\).
Step 1 ：Define the functions \(f(x) = x^{2} + 6\) and \(g(x) = \sqrt{x - 3}\).
Step 2 ：Find the composite function \(f \circ g\). This is done by substituting \(g(x)\) into \(f(x)\).
Step 3 ：Substitute \(g(x) = \sqrt{x - 3}\) into \(f(x) = x^{2} + 6\).
Step 4 ：Simplify the expression to get the composite function \(f \circ g\).
Step 5 ：The composite function \(f \circ g(x)\) simplifies to \(x + 3\). Therefore, \(f \circ g(x) = x + 3\) for \(x \geq 3\).
Step 6 ：Final Answer: \(\boxed{f \circ g(x) = x + 3}\) for \(x \geq 3\).
