From graph of the rational function, you must define $\displaystyle a, b, c, d$ in $\displaystyle f(x)=\frac{ax+b}{cx+d}$.

(I'm assuming both x-axisandy-axis are crossed at $\displaystyle -\frac{1}{2}$)

I start with this set-up: $\displaystyle \frac{b}{d}=-\frac{1}{2}$ (because of $\displaystyle f(0)=-\frac{1}{2}$)

$\displaystyle \frac{a}{c}=2$ (asymptote)

$\displaystyle -\frac{1}{2}a+b=0$ (because at $\displaystyle x=-\frac{1}{2}$ the function cross the x-axis and is equal to zero)

$\displaystyle 2c+d=0$ (becausepoleof the function is at $\displaystyle x=2$).

So it comes down to4 equations with 4 unknown variables, but I've tried many times solving, and I constantly get trapped inloop(you know, where your "new" equation is actually some old ones mixed up?)!!

Where is the flaw in my logic?