Question 1185294
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The conditions are over-specified; there is no rational function that satisfies all the conditions....<br>
(1) vertical asymptotes at x=3 and x=-3:<br>
This means there must be factor(s) of (x-3) and (x+3) in the denominator; and no other linear factors<br>
{{{r(x)=a/(((x-3)^m)((x+3)^n))}}}<br>
(2) horizontal asymptote at y=1:<br>
This means the leading terms of the numerator and denominator are the same (same coefficient and same power).  That means the coefficient a is 1, and the number of linear factors is the same in numerator and denominator.<br>
{{{r(x)=1/(((x-3)^m)((x+3)^n))}}}<br>
(3) only x-intercept at x=5:<br>
The only factor(s) in the numerator are (x-5).  From (2), the total number of linear factors in the numerator and denominator must be the same<br>
{{{r(x)=(x-5)^(m+n)/(((x-3)^m)((x+3)^n))}}}<br>
(4) y-intercept -5/9:<br>
Set x=0 and see what happens<br>
{{{r(0)=(-5)^(m+n)/(((-3)^m)((3)^n))=-5/9}}}<br>
Ignoring signs, with only factors of 5 in the numerator and only factors of 3 in the denominator, the only way to get a y-intercept of 5/9 is with one factor of 5 in the numerator and two factors of 3 in the denominator.  But that would make the function<br>
{{{r(x)=(x-5)/((x-3)(x+3))}}}<br>
And that has both the wrong y-intercept (5/9 instead of -5/9) and the wrong horizontal asymptote (y=0 instead of y=1).<br>
ANSWER: There is no rational function with all the prescribed conditions<br>