Answer
$f\circ g(x)=|x+4|-4 \qquad $ domain: $(-\infty,\infty)$
$ g\circ f(x)=|x|; \quad$ domain: $(-\infty,\infty)$
$ f\circ f(x)=x-8; \quad$ domain: $(-\infty,\infty)$
$ g\circ g(x)=|x+4|+4; \quad$ domain: $(-\infty,\infty)$
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)-4\quad$on the domain of $g$(x)
$=|x+4|-4 \qquad $ domain: $(-\infty,\infty)$
$g\circ f(x)=g[f(x)]=|f(x)+4| \quad$on the domain of f(x),\
$=|x-4+4| $
$=|x|; \quad$ domain: $(-\infty,\infty)$
$ f\circ f(x)=f[f(x)]=f(x)-4,\qquad$on the domain of f(x)
$=(x-4)-4,\quad $ on $(-\infty,\infty)$
$=x-8; \quad$ domain: $(-\infty,\infty)$
$ g\circ g(x)=g[g(x)]=|g(x)+4| \quad$on the domain of $g$(x)
$=||x+4|+4|\quad$on $(-\infty,\infty)$
... when a and b are both positive, then $|a+b|=a+b...$
$=|x+4|+4; \quad$ domain: $(-\infty,\infty)$
