Answer
$\text{a) }
f(g(4))=\dfrac{35}{3}
\\\\\text{b) }
g(f(4))=12$
Work Step by Step
$\bf{\text{Solution Outline:}}$
With
\begin{array}{l}\require{cancel}
f(x)=
\dfrac{2}{3}x+\dfrac{1}{3}
\\g(x)=
5x-3
,\end{array}
to find $
f(g(4))
,$ find first $
g(4)
.$ Then substitute the result in $f.$
To find $
g(f(4))
,$ find first $
f(4)
.$ Then substitute the result in $g.$
$\bf{\text{Solution Details:}}$
a) Replacing $x$ with $
4
$ in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=5x-3
\\\\
g(4)=5(4)-3
\\\\
g(4)=20-3
\\\\
g(4)=17
.\end{array}
Replacing $x$ with the result above in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=\dfrac{2}{3}x+\dfrac{1}{3}
\\\\
f(17)=\dfrac{2}{3}(17)+\dfrac{1}{3}
\\\\
f(17)=\dfrac{34}{3}+\dfrac{1}{3}
\\\\
f(17)=\dfrac{35}{3}
.\end{array}
Hence, $
f(g(4))=\dfrac{35}{3}
.$
b) Replacing $x$ with $
4
$ in $f$ results to
\begin{array}{l}\require{cancel}
f(x)=\dfrac{2}{3}x+\dfrac{1}{3}
\\\\
f(4)=\dfrac{2}{3}(4)+\dfrac{1}{3}
\\\\
f(4)=\dfrac{8}{3}+\dfrac{1}{3}
\\\\
f(4)=\dfrac{9}{3}
\\\\
f(4)=3
.\end{array}
Replacing $x$ with the result above in $g$ results to
\begin{array}{l}\require{cancel}
g(x)=5x-3
\\\\
g(3)=5(3)-3
\\\\
g(3)=15-3
\\\\
g(3)=12
.\end{array}
Hence, $
g(f(4))=12
.$
Therefore,
\begin{array}{l}\require{cancel}
\text{a) }
f(g(4))=\dfrac{35}{3}
\\\\\text{b) }
g(f(4))=12
.\end{array}