Answer
$(g\circ f)(-2)=2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the value of the given expression, $
\left( g\circ f \right)(-2)
,$ use the definition of function composition. Then use the values in the given table.
$\bf{\text{Solution Details:}}$
Using $(f\circ g)(x)=f(g(x)),$ then
\begin{array}{l}\require{cancel}
(g\circ f)(-2)=g(f(-2))
.\end{array}
Based on the given table, when $x=
-2
,$ the value of $f(x)$ is $
1
.$ Hence, $
f(-2)=1
.$
By substitution, the equation of the function composition above becomes
\begin{array}{l}\require{cancel}
(g\circ f)(-2)=g(f(-2))
\\\\
(g\circ f)(-2)=g(1)
.\end{array}
Based on the given table, when $x=
1
,$ the value of $g(x)$ is $
2
.$ Hence, $
g(1)=2
.$
By substitution, the equation of the function composition above becomes
\begin{array}{l}\require{cancel}
(g\circ f)(-2)=g(f(-2))
\\\\
(g\circ f)(-2)=g(1)
\\\\
(g\circ f)(-2)=2
.\end{array}
