Answer
$f\circ g(x)=x+2;\qquad $ domain: all reals, $(-\infty,\infty)$
$ g\circ f(x)=\sqrt[3]{x^{3}+2}; \quad$ domain: all reals, $(-\infty,\infty)$
$ f\circ f(x)=(x^{3}+2)^{3}+2 ;\quad$ domain: all reals, $(-\infty,\infty)$
$ g\circ g(x)=\sqrt[9]{x}; \quad$ domain: all reals, $(-\infty,\infty)$
Work Step by Step
f(x) is defined for all x,
g(x) is defined for all x
$f\circ g(x)=f[g(x)]=[g(x)]^{3}+2$
$=(\sqrt[3]{x})^{3}+2$
$=x+2;\qquad $ domain: all reals, $(-\infty,\infty)$
$g\circ f(x)=g[f(x)]=\sqrt[3]{f(x)}$
$=\sqrt[3]{x^{3}+2}; \quad$ domain: all reals, $(-\infty,\infty)$
$f\circ f(x)=f[f(x)]=[f(x)]^{3}+2$
$=(x^{3}+2)^{3}+2 ;\quad$ domain: all reals, $(-\infty,\infty)$
$g\circ g(x)=g[g(x)]=\sqrt[3]{g(x)}$
$=\sqrt[3]{=\sqrt[3]{x}}$
$=\sqrt[9]{x}; \quad$ domain: all reals, $(-\infty,\infty)$