1. ## Rational Functions

Write an equation of a rational function of the form:

f(x) = ax + b / cx + d;

whose graph has the following feautures:

x-intercept of 1/4

y-intercept of -1/2

vertical asymptote w/equation x = -2/3

horizontal asymptote w/equation y = 4/3

Thank you

Write an equation of a rational function of the form:

f(x) = ax + b / cx + d;

whose graph has the following feautures:

x-intercept of 1/4

y-intercept of -1/2

vertical asymptote w/equation x = -2/3

horizontal asymptote w/equation y = 4/3

Thank you
I suspect that you are asked to find a function of the form $\displaystyle f(x)=(ax+b)/(cx+d)$ and that you were just a bit lazy in leaving out the paranthesis around the denominator and numerator terms.
If so, the following system of equations can be used to determine a,b,c, and d:

$\displaystyle \begin{array}{lcrcl} \text{(1)}&\qquad & a\cdot\tfrac{1}{4}+b &=& 0\\[.2cm] \text{(2)} && \tfrac{b}{d} &=& -\tfrac{1}{2}\\[.2cm] \text{(3)} && c\cdot\left(-\tfrac{2}{3}\right) +d &=& 0\\[.2cm] \text{(4)} && \tfrac{a}{c} &=& \tfrac{4}{3} \end{array}$
This system has infinitenly many solutions, because it is possible to multiply numerator and denominator with the same constant $\displaystyle \neq 0$ to get another solution of the original problem. One, particularly nice solution (i.e. one with small, integral values for a,b,c, and d) is a=4, b=-1, c=3, d=2.

3. thank you

4. Hello everyone

Just a quick correction to a typo. I'm sure Failure intended equation (1) to read:
$\displaystyle a\cdot\tfrac14+b=0$