Consider the rational function f given below fx x3 5x2 6x

Consider the rational function f given below. f(x) = x^3 + 5x^2 + 6x/x^2 - 4x - 12 Find the domain of f. (Enter your answer using interval notation.) Identify any vertical asymptotes of the graph of y = f(x) (If an answer does not exist, enter DNE.) x = Identify any holes in the graph. (If an answer does not exist, enter DNE.) (x, y) = () Find the horizontal asymptote, if it exists. (If an answer does not exist, enter DNE.) y=

Solution

f(x) = (x^3 + 5x^2 +6x)/(x^2 -4x -12)

a) Domain: denominator should not be zero

x^2 -4x -12 =0

x^2 -6x +2x -12 =0 ----> x(x-6) +2(x-6) =0

(x+2)(x-6) =0

Domain : x= -2 , 6

So, ( -inf , -2)U( 6 , inf)

b) Vertical asymtotes : x= -2 ; x=6

c) Holes are points at which numerator and den. are zero:

f(x) = (x^3 + 5x^2 +6x)/(x^2 -4x -12)

= x(x^2 +5x +6)/(x+2)(x+6)

= x(x+2)(x+3) / (x+2)(x+6)

Hole at x = -2

d) if the degree of the numerator is exactly one more than the degree of the denominator (so that the polynomial fraction is \"improper\"), then the graph of the rational function will be, roughly, a slanty straight line

So, no horizontal asymtotes :

y = dne