[tex]\begin{gathered} f(g(x))\text{ = }\sqrt[]{x^2+9}\text{ + 6} \\ \\ g(f(x))\text{ = x }+12\text{ }\sqrt[]{x}\text{ + 45} \end{gathered}[/tex]Explanation:[tex]\begin{gathered} \text{Given:} \\ f(x)=\text{ }\sqrt[]{x}\text{ + 6} \\ g(x)=x^2+\text{ 9} \end{gathered}[/tex]
To get f(g(x)): we will substitute the x in f(x) with the function g(x):
[tex]\begin{gathered} f(g(x))\text{ = }\sqrt[]{(x^2+9)}\text{ + 6} \\ f(g(x))\text{ = }\sqrt[]{x^2+9}\text{ + 6} \\ \text{ (it can't be simplified further)} \end{gathered}[/tex]
To get g(f(x)): we will substitute the x in g(x) with function f(x):
[tex]\begin{gathered} g(f(x))\text{ = (}\sqrt[]{x}+6)^2\text{ + 9} \\ g(f(x))\text{ = (}\sqrt[]{x}+6)\text{(}\sqrt[]{x}+6)\text{ + 9} \\ g(f(x))\text{ = (}\sqrt[]{x})\text{(}\sqrt[]{x}+6)\text{ }+6\text{ (}\sqrt[]{x}+6)\text{ + 9} \end{gathered}[/tex][tex]\begin{gathered} g(f(x))\text{ = x }+6\text{ }\sqrt[]{x}\text{ }+6\text{ }\sqrt[]{x}\text{ + 36 + 9} \\ g(f(x))\text{ = x }+12\text{ }\sqrt[]{x}\text{ + 45} \end{gathered}[/tex]
