Question

give an example of a rational function whose graph has two vertical asymptotes, x=2 amd x=3. two x-intercepts, 0 and -3, and one horizontal asymptote, y=2

Answer 1

`\boxed{f(x)=(2x(x+3))/((x-2)(x-3))}\\\\check:\\vertical\ asymptotes:(x-2)(x-3)\\eq0\to x\\eq2\ and\ x\\eq3\\\boxed{x=2\ and\ x=3}\\\\x-intercepts:f(x)=0\iff(2x(x+3))/((x-2)(x-3))=0\iff2x(x+3)=0\\\iff 2x=0\ or\ x+3=0\iff \boxed{x=0\ or\ x=-3}\\\\horizontal\ asymptote:\\\lim\limits_(x\to\pm\infty)(2x(x+3))/((x-2)(x-3))=\lim\limits_(x\to\pm\infty)(2x^2+6x)/(x^2-5x+6)=\lim\limits_(x\to\pm\infty)(x^2(2+(6)/(x)))/(x^2(1-(5)/(x)+(6)/(x^2)))=(2)/(1)=\fbox2`