Question 1166001
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As we will see, there are too many constraints; there is no rational function that meets all the requirements.<br>
(a) Vertical asymptotes at x=3 and x=-3<br>
This requires factors of (x-3) and (x+3) (and no others) in the denominator:<br>
{{{f(x) = a/((x+3)(x-3))}}}<br>
(b) Only x-intercept at x=5<br>
This requires a factor of (x-5) in the numerator, and no other factors in the numerator except a constant:<br>
{{{f(x) = a(x-5)/((x+3)(x-3))}}}<br>
(c) Horizontal asymptote at y=1<br>
This requires that the degree of the numerator be equal to the degree of the denominator; it also requires that the constant be a=1.<br>
The only factors in the denominator are (x+3) and (x-3), and the only factor in the numerator is (x-5).  For the degrees of the numerator and denominator to be equal, the factor in the numerator needs to be there twice:<br>
{{{f(x) = (x-5)^2/((x+3)(x-3))}}}<br>
At this point, there are no unknowns in the function; it is completely determined by the horizontal and vertical asymptotes and the x-intercept.<br>
A graph showing the vertical asymptotes and the single x-intercept; the horizontal asymptote doesn't show up well because the function value gets close to 1 only for very large positive or very large negative values of x....<br>
{{{graph(400,400,-10,10,-10,10,(x-5)^2/((x+3)(x-3)))}}}<br>
(d) y-intercept -5/9<br>
This is problematic.  The function as it currently stands has a y-intercept of -25/9.<br>
If we were to introduce a factor of 1/5 in the function to get a y-intercept of -5/9, then the horizontal asymptote would be y=1/5 instead of y=1.<br>
So we can't find a rational function that satisfies all of the given conditions.<br>