Question 1169077
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Vertical asymptotes at x=3 and x=-3 --> factors of (x+3) and (x-3) in the denominator:<br>
{{{f(x) = a/((x+3)(x-3))}}}<br>
Only x-intercept at x=5 --> only linear factors in the numerator are (x-5):<br>
{{{f(x) = (a(x-5)^n)/((x+3)(x-3))}}}<br>
Horizontal asymptote at y=1 --> degrees of numerator and denominator are the same, and leading coefficients are the same.  So a=1; and the number of factors of (x-5) must be 2:<br>
{{{f(x) = ((x-5)^2)/((x+3)(x-3))}}}<br>
y-intercept 5/9....<br>
This is a problem; the given requirements are inconsistent.<br>
With the function we have at this point, the y-intercept is -25/9.<br>
If we add a constant factor to make the y-intercept 5/9, then we no longer have a horizontal asymptote at y=1.<br>
ANSWER: The given conditions are inconsistent; there is no rational function with all the given features.<br>