Answer
$\color{blue}{\bf{ (a) \sqrt{2x-2} ; [1,\infty) }}$
$\color{blue}{\bf{ (b) 2 \sqrt{x-2} ; [2,\infty) }}$
Work Step by Step
We are given the two functions $\bf{f}$ and $\bf{g}$
$\bf{f(x) = \sqrt{x-2} }$ and $\bf{g(x) = 2x }$
$\bf(a)$We are asked to find $\bf{ ( f \text{ }\omicron\text{ g} )( x ) }$ and its domain.
${ ( f \text{ }\omicron\text{ g} )( x ) }= \sqrt{2x-2} $
$\color{blue}{\bf{ \sqrt{2x-2} }}$
$2x-2\geq0$ or $ \sqrt{2x-2} $ would be imaginary
$2x\geq2$
$x\geq1$
$\color{blue}{\bf{ [1,\infty) }}$
$\bf(b)$We are asked to find $\bf{ ( g \text{ }\omicron\text{ f} )( x ) }$ and its domain.
${ ( g \text{ }\omicron\text{ f} )( x ) }= 2 \sqrt{x-2} $
$\color{blue}{\bf{ 2 \sqrt{x-2} }}$
$x-2\geq0$ or $ \sqrt{x-2} $ would be imaginary
$x\geq2$
$\color{blue}{\bf{ [2,\infty) }}$
