Answer
a) $|x+6|$; domain $(-\infty,\infty)$
b) $|x|+6$; domain: $(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=|x|$
$g(x)=x+6$
Determine the domains $D_f$ and $D_g$ of the two functions:
$D_f=(-\infty,\infty)$
$D_g=(-\infty,\infty)$
a) Find $f\circ g$ and its domain $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(x+6\right)=|x+6|$
$D_{f\circ g}=(-\infty,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(|x|)=|x|+6$
$D_{g\circ f}=(-\infty,\infty)$
