Answer
a) $\dfrac{1}{x+3}$; domain $(-\infty,-3)\cup(-3,\infty)$
b) $\dfrac{3x+1}{x}$; domain: $(-\infty,0)\cup(0,\infty)$
Work Step by Step
We are given the functions:
$f(x)=\dfrac{1}{x}$
$g(x)=x+3$
Determine the domains $D_f$ and $D_g$ of the two functions:
$D_f=(-\infty,0)\cup(0,\infty)$
$D_g=(-\infty,\infty)$
a) Find $f\circ g$ and its domain $D_{f\circ g}$:
$(f\circ g)(x)=f(g(x))=f\left(x+3\right)=\dfrac{1}{x+3}$
$x+3=0\Rightarrow x=-3$
$D_{f\circ g}=(-\infty,-3)\cup(-3,\infty)$
b) Find $g\circ f$ and its domain $D_{g\circ f}$:
$(g\circ f)(x)=g(f(x))=g\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+3=\dfrac{3x+1}{x}$
$D_{g\circ f}=(-\infty,0)\cup(0,\infty)$
