Answer
$(g\circ f)(6)=12$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
(g\circ f)(6)
,$ use the definition of function composition. Use the given table for the values of the function.
$\bf{\text{Solution Details:}}$
Since $(f\circ g)(x)=f(g(x)),$ then
\begin{array}{l}\require{cancel}
(g\circ f)(6)=g(f(6))
.\end{array}
Based on the table, the value of $
f
$ when $x=
6
$ is $
9
.$ Hence, $
f(6)=9
.$ By substitution, the equation above becomes
\begin{array}{l}\require{cancel}
(g\circ f)(6)=g(f(6))
\\\\
(g\circ f)(6)=g(9)
.\end{array}
Based on the table, the value of $
g
$ when $x=
9
$ is $
12
.$ Hence, $
g(9)=12
.$ By substitution, the equation above becomes
\begin{array}{l}\require{cancel}
(g\circ f)(6)=g(f(6))
\\\\
(g\circ f)(6)=g(9)
\\\\
(g\circ f)(6)=12
.\end{array}