Answer
$ a.\qquad (f\circ g)(x)=x^{2}+5,\qquad$ Domain: $\mathbb{R}$
$ b.\qquad (g\circ f)(x)=x^{2}+2x+5,\qquad$ Domain: $\mathbb{R}$
$ c.\qquad (f\circ f)(x)=x+2,\qquad$ Domain: $\mathbb{R}$
$ d.\qquad(g\circ g)(x)=x^{4}+8x^{2}+20,\qquad$ Domain: $\mathbb{R}$
Work Step by Step
$ f(x)=x+1,\qquad$ Domain: $\mathbb{R}$
$ g(x)=x^{2}+4,\qquad$ Domain: $\mathbb{R}$
$a.$
$f\circ g(x)=f[g(x)]=g(x)+1$
$=x^{2}+4+1$
$=x^{2}+5,\qquad$ Domain: $\mathbb{R}$
$b.$
$(g\circ f)(x)=g[f(x)] =[f(x)]^{2}+4$
$=(x+1)^{2}+4$
$=x^{2}+2x+1+4$
$=x^{2}+2x+5,\qquad$ Domain: $\mathbb{R}$
$c.$
$f\circ f(x)=f[f(x)]=f(x)+1$
$=x+1+1$
$=x+2,\qquad$ Domain: $\mathbb{R}$
$d.$
$(g\circ g)(x)=g[g(x)] =[g(x)]^{2}+4$
$=(x^{2}+4)^{2}+4$
$=x^{4}+8x^{2}+16+4$
$=x^{4}+8x^{2}+20,\qquad$ Domain: $\mathbb{R}$
