Answer
$f(x)=\dfrac{3}{4}x+\dfrac{7}{2}$
Work Step by Step
Using the properties of equality, the given equation, $ 4x+3y=5 ,$ is equivalent to \begin{array}{l} 3y=-4x+5 \\\\ y=-\dfrac{4}{3}x+\dfrac{5}{3} .\end{array} Using $y=mx+b$, where $m$ is the slope, the slope of the given line is \begin{array}{l} m=-\dfrac{4}{3} .\end{array} Using $m= \dfrac{3}{4} $ (negative reciprocal slope since the lines are perpendicular) and the given point $( -6,-1 ),$ then the equation of the line is \begin{array}{l} y-(-1)=\dfrac{3}{4}(x-(-6)) \\\\
y+1=\dfrac{3}{4}(x+6)
\\\\
y+1=\dfrac{3}{4}x+\dfrac{18}{4}
\\\\
y=\dfrac{3}{4}x+\dfrac{18}{4}-1
\\\\
y=\dfrac{3}{4}x+\dfrac{18}{4}-\dfrac{4}{4}
\\\\
y=\dfrac{3}{4}x+\dfrac{14}{4}
\\\\
y=\dfrac{3}{4}x+\dfrac{7}{2}
.\end{array}
In function notation, this is equivalent to $
f(x)=\dfrac{3}{4}x+\dfrac{7}{2}
.$
