Answer
$\color{blue}{\bf{ (a) \sqrt{x-1} ; [1,\infty) }}$
$\color{blue}{\bf{ (b) \sqrt{x}-1 ; [0,\infty) }}$
Work Step by Step
We are given the two functions $\bf{f}$ and $\bf{g}$
$\bf{f(x) = \sqrt{x} }$ and $\bf{g(x) = x -1 }$
$\bf(a)$We are asked to find $\bf{ ( f \text{ }\omicron\text{ g} )( x ) }$ and its domain.
${ ( f \text{ }\omicron\text{ g} )( x ) }= \sqrt{x-1} $
$\color{blue}{\bf{ \sqrt{x-1} }}$
$x-1\geq0$ or else $ \sqrt{x-1} $ would be imaginary
$x\geq1$ so the domain is:
$\color{blue}{\bf{ [1,\infty) }}$
$\bf(b)$We are asked to find $\bf{ ( g \text{ }\omicron\text{ f} )( x ) }$ and its domain.
${ ( g \text{ }\omicron\text{ f} )( x ) }= \sqrt{x}-1 $
$\color{blue}{\bf{ \sqrt{x}-1 }}$
$x\geq0$ or else $ \sqrt{x} $ would be imaginary, so the domain is:
$\color{blue}{\bf{ [0,\infty) }}$
