[tex]\begin{gathered} (f\circ g)(x)\text{ }=\frac{x}{9} \\ (g\circ f)(x)=9x-48 \end{gathered}[/tex]
Explanation
A composite function is a function that is written inside another function. Composition of a function is done by substituting one function into another function.
f [g (x)] is the composite function of f (x) and g (x). so
[tex]\begin{gathered} f(x)\text{ }=\frac{1}{x-6} \\ g(x)=\frac{9}{x}+6 \end{gathered}[/tex]
hence
Step 1
[tex](f\circ g)(x)[/tex]
so
[tex]\begin{gathered} f(x)\text{ }=\frac{1}{x-6} \\ (f\circ g)(x)\text{ }=\frac{1}{g(x)-6}=\frac{1}{(\frac{9}{x}+6)-6}=\frac{1}{\frac{9}{x}}=\frac{x}{9} \\ (f\circ g)(x)\text{ }=\frac{x}{9} \end{gathered}[/tex]
Step 2
[tex]\begin{gathered} (g\circ f)(x) \\ so \\ (g\circ f)(x)=\frac{9}{f(x)}+6=\frac{9}{\frac{1}{x-6}}+6=\frac{9(x-6)}{1}+6=9x-54+6=9x-48 \\ (g\circ f)(x)=9x-48 \end{gathered}[/tex]
I hope this helps you
