Answer
$\text{a) }
f(g(x))=\dfrac{10}{3}x-\dfrac{5}{3}
\\\\\text{b) }
g(f(x))=\dfrac{4}{3}x+\dfrac{1}{15}$
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}
replace $x$ with $g(x)$ in $f$ to find $f(g(x)).$ To find $g(f(x)),$ replace $x$ with $f(x)$ in $g.$
$\bf{\text{Solution Details:}}$
Replacing $x$ with $g(x)$ in $f,$ then
\begin{array}{l}\require{cancel}
f(g(x))=f\left( 5x-3 \right)
\\\\
f(g(x))=\dfrac{2}{3}\left( 5x-3 \right)+\dfrac{1}{3}
\\\\
f(g(x))=\dfrac{10}{3}x-2+\dfrac{1}{3}
\\\\
f(g(x))=\dfrac{10}{3}x-\dfrac{6}{3}+\dfrac{1}{3}
\\\\
f(g(x))=\dfrac{10}{3}x-\dfrac{5}{3}
.\end{array}
Replacing $x$ with $f(x)$ in $g.$ Hence,
\begin{array}{l}\require{cancel}
g(f(x))=g\left( \dfrac{2}{3}x+\dfrac{1}{3} \right)
\\\\
g(f(x))=2\left( \dfrac{2}{3}x+\dfrac{1}{3} \right)-\dfrac{3}{5}
\\\\
g(f(x))=\dfrac{4}{3}x+\dfrac{2}{3} -\dfrac{3}{5}
\\\\
g(f(x))=\dfrac{4}{3}x+\dfrac{10}{15} -\dfrac{9}{15}
\\\\
g(f(x))=\dfrac{4}{3}x+\dfrac{1}{15}
.\end{array}
Hence,
\begin{array}{l}\require{cancel}
\text{a) }
f(g(x))=\dfrac{10}{3}x-\dfrac{5}{3}
\\\\\text{b) }
g(f(x))=\dfrac{4}{3}x+\dfrac{1}{15}
.\end{array}
