Answer
$ f(x)=-\frac{2}{3}x-2$.
Work Step by Step
The given perpendicular line equation is
$\Rightarrow 3x-2y=4$
The slope-intercept form is
$\Rightarrow y=\frac{3}{2}x-2$
Slope of the above line is $m_1=\frac{3}{2}$
and $y-$intercept is $-2$
Two lines are perpendicular if their slopes are negative reciprocal to each other.
Hence, slope of the perpendicular line
$\Rightarrow m_2=−\frac{1}{m_1}$
$\Rightarrow m_2=−\frac{1}{\frac{3}{2}}$
Simplify.
$\Rightarrow m_2=−\frac{2}{3}$
$y-$ intercept is same for both lines
Hence, the $y-$ intercept for the required line is $-2$.
The standard slope-intercept form is
$\Rightarrow y=mx+c$
Plug all values.
$\Rightarrow y=(-\frac{2}{3})x-2$
Plug $y=f(x)$.
$\Rightarrow f(x)=-\frac{2}{3}x-2$.
