Use the rule of correspondence of f and g as well as the definition of the composition of two functions to find the requested expressions.
Remember that given two functions, f₁ and f₂:
[tex](f_1\circ f_2)(x)=f_1(f_2(x))[/tex]
Since:
[tex]\begin{gathered} f(x)=\frac{1}{x} \\ g(x)=2x+4 \end{gathered}[/tex]
Then:
[tex]\begin{gathered} (f\circ f)(x)=f(f(x)) \\ =\frac{1}{f(x)} \\ =\frac{1}{(\frac{1}{x})} \\ =x \end{gathered}[/tex]
The domain of f(f(x)) must match the domain of f(x):
[tex]D_{f\circ f}=D_f=(-\infty,0)\cup(0,\infty)[/tex]
On the other hand:
[tex]\begin{gathered} (g\circ g)(x)=g(g(x)) \\ =2\cdot g(x)+4 \\ =2(2x+4)+4 \\ =4x+8+4 \\ =4x+12 \end{gathered}[/tex]
The domain is the same as the domain of g:
[tex](-\infty,\infty)[/tex]
Therefore, the answers are:
[tex]\begin{gathered} (f\circ f)(x)=x \\ D_{f\circ f}=(-\infty,0)\cup(0,\infty)_{} \end{gathered}[/tex][tex]\begin{gathered} (g\circ g)(x)=4x+12 \\ D_{g\circ g}=(-\infty,\infty) \end{gathered}[/tex]
