Answer
$f(x)=\dfrac{1}{4}x-\dfrac{7}{2}$
Work Step by Step
Using $y=mx+b$ where $m$ is the slope, the given equation, $
4x+y=\dfrac{2}{3}
,$ is equivalent to
\begin{array}{l}\require{cancel}
y=-4x+\dfrac{2}{3}
.\end{array}
Hence, the slope is $m=-4$. Since perpendicular lines have negative reciprocal slopes, then $m_p=\dfrac{1}{4}$
Using $
(2,-3)
$ and $m_p=
\dfrac{1}{4}
,$ the equation of the line is
\begin{array}{l}\require{cancel}
y-(-3)=\dfrac{1}{4}(x-2)
\\\\
y+3=\dfrac{1}{4}(x-2)
\\\\
y+3=\dfrac{1}{4}x-\dfrac{1}{2}
\\\\
y=\dfrac{1}{4}x-\dfrac{1}{2}-3
\\\\
y=\dfrac{1}{4}x-\dfrac{1}{2}-\dfrac{6}{2}
\\\\
y=\dfrac{1}{4}x-\dfrac{7}{2}
.\end{array}
In function notation form, this is equivalent to $
f(x)=\dfrac{1}{4}x-\dfrac{7}{2}
.$