Answer
$\text{a) }
f(g(5))=-55
\\\\\text{b) }
g(f(5))=-129$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
3x+8
\\g(x)=
-6x+9
,\end{array}
to find $
f(g(5))
,$ find first $
g(5)
.$ Then substitute the result in $f.$
To find $
g(f(5))
,$ find first $
f(5)
.$ Then substitute the result in $g.$
$\bf{\text{Solution Details:}}$
a) Replacing $x$ with $
5
$ in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=-6x+9
\\\\
g(5)=-6(5)+9
\\\\
g(5)=-30+9
\\\\
g(5)=-21
.\end{array}
Replacing $x$ with the result above in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=3x+8
\\\\
f(-21)=3(-21)+8
\\\\
f(-21)=-63+8
\\\\
f(-21)=-55
.\end{array}
Hence, $
f(g(5))=-55
.$
b) Replacing $x$ with $
5
$ in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=3x+8
\\\\
f(5)=3(5)+8
\\\\
f(5)=15+8
\\\\
f(5)=23
.\end{array}
Replacing $x$ with the result above in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=-6x+9
\\\\
g(23)=-6(23)+9
\\\\
g(23)=-138+9
\\\\
g(23)=-129
.\end{array}
Hence, $
g(f(5))=-129
.$
Therefore,
\begin{array}{l}\require{cancel}
\text{a) }
f(g(5))=-55
\\\\\text{b) }
g(f(5))=-129
.\end{array}
