Answer
x-intercept: (-2,0) (1,0)
y-intercept: (0, 1/6)
Vertical Asymptote: x=-4, 3
Domain: (-∞, -4) U (-4, 3) U (3, ∞)
Horizontal Asymptote: y=1
Range: (-∞, 1) U (1, ∞)
See graph below
Work Step by Step
$r(x)=\frac{(x-1)(x+2)}{(x+4)(x-3)}$
$(x-1)(x+2) = 0$
x = 1, -2
x-intercept: (1,0) (-2, 0)
y-intercept is the ratio of the constants, which is -2/ -12 = 1/6
Thus, the y-intercept is at (0, 1/6)
Vertical asymptotes are when the denominator is equal to 0
$(x+4)(x-3) = 0$
x = -4, 3
So, domain is from (-∞, -4) U (-4, 3) U (3, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
1 / 1 = 1
Thus, the horizontal asymptote is at y=1
So the range is from (-∞, 1) U (1, ∞)
