Answer
$(g\circ f)(3)=1$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the value of the given expression, $
\left( g\circ f \right)(3)
,$ given that
\begin{array}{l}\require{cancel}
f(x)=\sqrt{x-2}
\\g(x)=x^2
,\end{array}
use the definition of function composition.
$\bf{\text{Solution Details:}}$
Using $(f\circ g)(x)=f(g(x)),$ then
\begin{array}{l}\require{cancel}
(g\circ f)(3)=g(f(3))
.\end{array}
Substituting $x$ with $3$ in $f(x)=\sqrt{x-2},$ then
\begin{array}{l}\require{cancel}
f(3)=\sqrt{3-2}
\\\\
f(3)=\sqrt{1}
\\\\
f(3)=1
.\end{array}
Using $f(3)=1,$ the equation of the function composition above becomes
\begin{array}{l}\require{cancel}
(g\circ f)(3)=g(f(3))
\\\\
(g\circ f)(3)=g(1)
.\end{array}
Substituting $x$ with $1$ in $g(x)=x^2,$ then
\begin{array}{l}\require{cancel}
g(1)=1^2
\\\\
g(1)=1
.\end{array}
Using $g(1)=1,$ the equation of the function composition above becomes
\begin{array}{l}\require{cancel}
(g\circ f)(3)=g(f(3))
\\\\
(g\circ f)(3)=g(1)
\\\\
(g\circ f)(3)=1
.\end{array}