Recall that:
[tex]\begin{gathered} f(g(x))=f(x^2+3), \\ g(f(x))=g(\sqrt{x}+8). \end{gathered}[/tex]Evaluating the functions, we get:
[tex]\begin{gathered} f(g(x))=\sqrt{x^2+3}+8, \\ g(f(x))=(\sqrt{x}+8)^2+3. \end{gathered}[/tex]Simplifying the second function, we get:
[tex]g(f(x))=x+16\sqrt{x}+64+3=x+16\sqrt{x}+67.[/tex]Answer:
[tex]\begin{gathered} f(g(x))=\sqrt{x^2+3}+8, \\ g(f(x))=x+16\sqrt{x}+67. \end{gathered}[/tex]