Answer
a) $|-x-1|$; domain $(-\infty,\infty)$
b) $3-|x-4|$; domain: $(-\infty,\infty)$
Work Step by Step
We are given the functions:
$f(x)=|x-4|$
$g(x)=3-x$
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 $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(3-x\right)=|3-x-4|=|-x-1|$
$D_{f\circ g}=(-\infty,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g(|x-4|)=3-|x-4|$
$D_{g\circ f}=(-\infty,\infty)$
