Answer
$B=-1$
Work Step by Step
We solve:
$f(x)=\displaystyle \frac{2x-B}{3x+4}$ and $f(2)=\displaystyle \frac{1}{2}$
We solve for $B$:
$f(2)=\frac{2*2-B}{3*2+4}$
$f(2)=\frac{4-B}{10}$
Since $f(2)=1/2$:
$\frac{1}{2}=\frac{4-B}{10}$
$5=4-B$
$B=-1$