Answer
x-intercept: (-2,0) (3,0)
y-intercept: None
Vertical Asymptote: x=-3, 0
Domain: (-∞, -3) U (-3, 0) U (0, ∞)
Horizontal Asymptote: y=1
Range: (-∞, ∞)
See graph below
Work Step by Step
$r(x)=\frac{x^2 - x - 6}{(x^2 + 3x)}$
$x^2 - x - 6 = 0$
$(x-3) (x+2) = 0$
x = -2, 3
x-intercept: (-2,0) (3,0)
No y-intercept as when x=0, the function is undefined
Vertical asymptotes are when the denominator is equal to 0
$x^2 + 3x = 0$
$x(x+3) = 0$
x = -3, 0
So, domain is from (-∞, -3) U (-3, 0) U (0, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
1/1 = 2
Thus, the horizontal asymptote is at y=1
So the range is from (-∞, ∞)
