Question

for $f(x) = x^2 + 8$ and $g(x) = \sqrt{x - 2}$, find the following composite functions and state the domain of each.\
(a) $f \circ g$ \quad (b) $g \circ f$ \quad (c) $f \circ f$ \quad (d) $g \circ g$\
\
\bigcirc a. the domain of $g \circ f$ is $\\{x \mid \\ \\}$.\
\quad (type an inequality. simplify your answer. use integers or fractions for any numbers in\
\quad the expression.)\
\bigcirc b. the domain of $g \circ f$ is all real numbers.\
\
(c) $(f \circ f)(x) = x^4 + 16x^2 + 72$ (simplify your answer.)\
select the correct choice below and fill in any answer boxes within your choice.\
\bigcirc a. the domain of $f \circ f$ is $\\{x \mid \\ \\}$.\
\quad (type an inequality. simplify your answer. use integers or fractions for any numbers in\
\quad the expression.)\
\bigcirc b. the domain of $f \circ f$ is all real numbers.\
\
(d) $(g \circ g)(x) = \square$ (simplify your answer.)

Explanation:

Step1: Recall Composite Function Rule

To find \( (g \circ g)(x) \), we substitute \( g(x) \) into itself. So \( (g \circ g)(x)=g(g(x)) \).

Step2: Substitute \( g(x) \) into \( g \)

Given \( g(x)=\sqrt{x - 2} \), then \( g(g(x))=g(\sqrt{x - 2})=\sqrt{\sqrt{x - 2}-2} \).

Step3: Determine the Domain (for Completeness, though the Question asks for Simplification)

But the question here is to simplify \( (g \circ g)(x) \), so we just need to write the composite function. The simplification of \( g(g(x)) \) is \( \sqrt{\sqrt{x - 2}-2} \).

Answer:

\(\sqrt{\sqrt{x - 2}-2}\)