Answer
$\text{a) }
f(g(3))=41
\\\\\text{b) }
g(f(3))=22$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
4x-7
\\g(x)=
x^2-3x+12
,\end{array}
to find $
f(g(3))
,$ find first $
g(3)
.$ Then substitute the result in $f.$
To find $
g(f(3))
,$ find first $
f(3)
.$ Then substitute the result in $g.$
$\bf{\text{Solution Details:}}$
a) Replacing $x$ with $
3
$ in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=x^2-3x+12
\\\\
g(3)=(3)^2-3(3)+12
\\\\
g(3)=9-9+12
\\\\
g(3)=12
.\end{array}
Replacing $x$ with the result above in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=4x-7
\\\\
f(12)=4(12)-7
\\\\
f(12)=48-7
\\\\
f(12)=41
.\end{array}
Hence, $
f(g(3))=41
.$
b) Replacing $x$ with $
3
$ in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=4x-7
\\\\
f(3)=4(3)-7
\\\\
f(3)=12-7
\\\\
f(3)=5
.\end{array}
Replacing $x$ with the result above in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=x^2-3x+12
\\\\
g(5)=(5)^2-3(5)+12
\\\\
g(5)=25-15+12
\\\\
g(5)=22
.\end{array}
Hence, $
g(f(3))=22
.$
Therefore,
\begin{array}{l}\require{cancel}
\text{a) }
f(g(3))=41
\\\\\text{b) }
g(f(3))=22
.\end{array}
