What are the vertical and horizontal asymptotes for the function mc016-1.jpg? horizontal asymptote: y = –2, y = 2 vertical asymptote: x = 3 horizontal asymptote: y = –4, y = 1 vertical asymptote: x = 3 horizontal asymptote: y = 3 vertical asymptote: x = –4, x = 1 horizontal asymptote: y = 3 vertical asymptote: x = –2, x = 2

F(x)=(3x²)/(x²-4)

Function is undefined when x²-4=0⇒when x=2 and x=-2

y=(3x²)/(x²-4), multiply both sides by x²-4
y*(x²-4)=(3x²*(x²-4))/(x²-4)
y*(x²-4)=3x²
The range is the set of y for which the discriminant is greater or equal to zero.
y*x²-4y=3x²
y*x²-4y-3x²=0
(y-3)*x²-4y=0
For the quadratic equation of the form ax²+bx+c=0, the discriminant is b²-4ac.
For this equation: a=y-3, b=0 and c=-4y ⇒
discriminant D is: 0²-4*(y-3)*(-4y)=4y*4*(y-3)
D=16y(y-3)=16y²-48y
16y²-48y≥0, by dividing both sides by 16⇒
y²-3y≥0
y(y-3)≥0⇒Function is undefined when y=0 and y=3

For this function, horizontal asymptote is y = 3 and vertical asymptotes are   x = –2, x = 2.