Question 1143054
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Write the function with both numerator and denominator as the product of linear factors.<br>
(1) vertical asymptotes at x=3 and x=-3: this means factors of (x-3) and (x+3) in the denominator.<br>
(2) horizontal asymptote y=1: this means the numerator and denominator have the same number of linear factors; and the leading coefficients are the same<br>
(3) only x-intercept is 5: this means the only factor in the numerator is (x-5).  However, since the numerator and denominator have to have the same number of factors, the numerator needs to have two factors of (x-5).<br>
We know the factors that are required in both numerator and denominator; and we know the leading coefficients in the numerator and denominator have to be the same.  That completely determines the function:<br>
{{{y = ((x-5)^2)/((x-3)(x+3))}}}<br>
However, the y-intercept for this function (found by setting x=0) is 25/-9 = -25/9; the problem states that the y-intercept is -5/9.<br>
We could get a y-intercept of -5/9 by adding a factor of 5 in the denominator:<br>
{{{y = ((x-5)^2)/(5((x-3)(x+3)))}}}<br>
But then the horizontal asymptote would be y=1/5 -- not y=1, as required.<br>
Alternatively, note that if the function were, instead,<br>
{{{y = -(x-5)/((x-3)(x+3))}}}<br>
then the y-intercept would be -5/9, as required.  But then the horizontal asymptote would be y=0, because the degree of the numerator is less than the degree of the denominator.<br>
So, in summary, there are too many requirements specified for the function; there is no rational function that meets all of the specified conditions.