Use the pair of functions to find $f(g(x))$ and $g(f(x))$. Simplify your answers.
\[
\begin{array}{l}
f(x)=x^{2}+8, \quad g(x)=\sqrt{x+4} \\
f(g(x))=\square \\
g(f(x))=\square
\end{array}
\]
Answer
So, the simplified forms of \(f(g(x))\) and \(g(f(x))\) are \(f(g(x)) = \boxed{x + 12}\) and \(g(f(x)) = \boxed{\sqrt{x^{2} + 12}}\).
Step 1 ：Given the functions \(f(x) = x^{2} + 8\) and \(g(x) = \sqrt{x + 4}\), we are asked to find \(f(g(x))\) and \(g(f(x))\).
Step 2 ：To find \(f(g(x))\), we substitute \(g(x)\) into \(f(x)\), so wherever we see \(x\) in \(f(x)\), we replace it with \(g(x)\).
Step 3 ：Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = (\sqrt{x + 4})^{2} + 8\).
Step 4 ：Simplifying this, we get \(f(g(x)) = x + 12\).
Step 5 ：To find \(g(f(x))\), we substitute \(f(x)\) into \(g(x)\), so wherever we see \(x\) in \(g(x)\), we replace it with \(f(x)\).
Step 6 ：Substituting \(f(x)\) into \(g(x)\), we get \(g(f(x)) = \sqrt{(x^{2} + 8) + 4}\).
Step 7 ：Simplifying this, we get \(g(f(x)) = \sqrt{x^{2} + 12}\).
Step 8 ：So, the simplified forms of \(f(g(x))\) and \(g(f(x))\) are \(f(g(x)) = \boxed{x + 12}\) and \(g(f(x)) = \boxed{\sqrt{x^{2} + 12}}\).
