Answer
$(f\circ g)(-6)=\sqrt{34}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the value of the given expression, $
\left( f\circ g \right)(-6)
,$ 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}
(f\circ g)(-6)=f(g(-6))
.\end{array}
Substituting $x$ with $-6$ in $g(x)=x^2,$ then
\begin{array}{l}\require{cancel}
g(-6)=(-6)^2
\\\\
g(-6)=36
.\end{array}
Using $g(-6)=36,$ the equation of the function composition above becomes
\begin{array}{l}\require{cancel}
(f\circ g)(-6)=f(g(-6))
\\\\
(f\circ g)(-6)=f(36)
.\end{array}
Substituting $x$ with $36$ in $f(x)=\sqrt{x-2},$ then
\begin{array}{l}\require{cancel}
f(36)=\sqrt{36-2}
\\\\
f(36)=\sqrt{34}
.\end{array}
Using $
f(36)=\sqrt{34}
,$ the equation of the function composition above becomes
\begin{array}{l}\require{cancel}
(f\circ g)(-6)=f(g(-6))
\\\\
(f\circ g)(-6)=f(36)
\\\\
(f\circ g)(-6)=\sqrt{34}
.\end{array}