Answer
$\color{blue}{\bf{(a) \dfrac{4}{ x+4} ; (-\infty, -4 )\bigcup( -4 ,\infty) }}$
$\color{blue}{\bf{(b) \dfrac{4}{x} +4 ; (-\infty, 0 )\bigcup( 0 ,\infty) }}$
Work Step by Step
We are given the two functions $\bf{f}$ and $\bf{g}$
$\bf{f(x) = \dfrac{4}{x} }$ and $\bf{g(x) = x+4 }$
${\bf(a)}$ We are asked to find $\bf{ ( f \text{ }\omicron\text{ g} )(x) }$ and its domain.
$( f \text{ }\omicron\text{ g} )(x) = \color{blue}{\bf{ \dfrac{4}{ x+4} }}$
$ x+4 \neq0$ because division by zero is undefined
$ x \neq-4$
so the domain is:
$\color{blue}{\bf(-\infty, -4 )\bigcup( -4 ,\infty) }$
${\bf(b)}$ We are asked to find $\bf{ ( g \text{ }\omicron\text{ f} )(x) }$ and its domain.
$( g \text{ }\omicron\text{ f} )(x) =\color{blue}{\bf{ \dfrac{4}{x} +4 }}$
$ x \neq0$ because division by zero is undefined so the domain is:
$\color{blue}{\bf(-\infty, 0 )\bigcup( 0 ,\infty) }$
