Answer
$f\circ g(x)=(x+1)^{2};\qquad $ domain: all reals, $(-\infty,\infty)$
$ g\circ f(x)=x^{2}+1; \quad$ domain: all reals, $(-\infty,\infty)$
$ f\circ f(x)=x^{4} ;\quad$ domain: all reals, $(-\infty,\infty)$
$ g\circ g(x)=x+2; \quad$ domain: all reals, $(-\infty,\infty)$
Work Step by Step
f(x) is defined for all x,
g(x) is defined for all x
$f\circ g(x)=f[g(x)]=[g(x)]^{2}$
$=(x+1)^{2};\qquad $ domain: all reals, $(-\infty,\infty)$
$g\circ f(x)=g[f(x)]=f(x)=1$
$=x^{2}+1; \quad$ domain: all reals, $(-\infty,\infty)$
$f\circ f(x)=f[f(x)]==[f(x)]^{2}$
$=(x^{2})^{2}$
$=x^{4} ;\quad$ domain: all reals, $(-\infty,\infty)$
$g\circ g(x)=g[g(x)]=g(x)+1$
$=(x+1)+1$
$=x+2; \quad$ domain: all reals, $(-\infty,\infty)$