Question 808180
Since {{{f}}} has a vertical is at {{{x = 1}}}, then the denominator of the rational function contains the term {{{(x - 1)}}}. Function {{{f}}} has the form.

{{{f(x) = g(x) / (x - 1)}}}

{{{g(x)}}} which is in the numerator must be of the same degree as the denominator since {{{f}}} has a horizontal asymptote. Also {{{g(x)}}} must contain the term {{{(x -3)}}} since {{{f}}} has a zero at {{{x = 3}}}. Hence

{{{f(x) =2 (x -3) / (x - 1)}}} 


{{{ drawing(600,600,   -10, 10, -10, 10, circle(0,6,0.2),circle(3,0,0.2), blue(line(1,10,1,10)), grid(0),graph(600, 600, -10, 10, -10, 10, 2(x -3)/(x - 1),2)) }}}