Answer
a) $\sqrt{x^2+4}$; domain $(-\infty,\infty)$
b) $x+4$; domain: $[-4,\infty)$
Work Step by Step
We are given the functions:
$f(x)=\sqrt{x+4}$
$g(x)=x^2$
Determine the domains $D_f$ and $D_g$ of the two functions:
$x+4\geq 0\Rightarrow x\geq -4$
$D_f=[-4,\infty)$
$D_g=(-\infty,\infty)$
a) Find $f\circ g$ and its domain $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(x^2\right)=\sqrt{x^2+4}$
$D_{f\circ g}=(-\infty,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(\sqrt{x+4})=(\sqrt{x+4})^2=x+4$
$D_{g\circ f}=[-4,\infty)$
