Answer
$(g\circ f)(-1) \text{ is undefined}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the value of the given expression, $
\left( g\circ f \right)(-1)
,$ 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)(-1)=g(f(-1))
.\end{array}
Substituting $x$ with $-1$ in $f(x)=\sqrt{x-2},$ then
\begin{array}{l}\require{cancel}
f(-1)=\sqrt{-1-2}
\\\\
f(-1)=\sqrt{-3}
\\\\
f(-1)=\text{not a real number}
.\end{array}
Since the function composition is only defined for real numbers, then $f(-1)$ is undefined and $
(g\circ f)(-1) \text{ is undefined}
.$
