College Algebra, 2013a

316 Rational Functions
4.1.2 Answers
1. f(x) = x
3x − 6
Domain: (−∞, 2) ∪ (2, ∞)
Vertical asymptote: x =2
As x → 2 − ,f(x) →−∞
As x → 2 + ,f(x) →∞
No holes in the graph
Horizontal asymptote: y = 1 3
As x →−∞,f(x) → 1 −
3
As x →∞,f(x) → 1 +
3
2. f(x) = 3+7x
5 − 2x
Domain: (−∞, 5 2 ) ∪ ( 5 2 , ∞)
Vertical asymptote: x = 5 2
As x → 5 −
2 ,f(x) →∞
As x → 5 +
2 ,f(x) →−∞
No holes in the graph
Horizontal asymptote: y = − 7 2
As x →−∞,f(x) →− 7 +
2
As x →∞,f(x) →− 7 −
2
x
3. f(x) =
x 2 + x − 12 = x
4. f(x) = x
(x +4)(x − 3)
x 2 +1
Domain: (−∞, −4) ∪ (−4, 3) ∪ (3, ∞)
Domain: (−∞, ∞)
Vertical asymptotes: x = −4,x=3
No vertical asymptotes
As x →−4 − ,f(x) →−∞
No holes in the graph
As x →−4 + ,f(x) →∞
Horizontal asymptote: y =0
As x → 3 − ,f(x) →−∞
As x →−∞,f(x) → 0 −
As x → 3 + ,f(x) →∞
As x →∞,f(x) → 0 +
No holes in the graph
Horizontal asymptote: y =0
As x →−∞,f(x) → 0 −
As x →∞,f(x) → 0 +
5. f(x) = x +7
(x +3) 2
6. f(x) = x3 +1
x 2 − 1 = x2 − x +1
x − 1
Domain: (−∞, −3) ∪ (−3, ∞)
Domain: (−∞, −1) ∪ (−1, 1) ∪ (1, ∞)
Vertical asymptote: x = −3
Vertical asymptote: x =1
As x →−3 − ,f(x) →∞
As x → 1 − ,f(x) →−∞
As x →−3 + ,f(x) →∞
As x → 1 + ,f(x) →∞
No holes in the graph
Hole at (−1, − 3 2 )
Horizontal asymptote: y =0
Slant asymptote: y = x
19 As x →−∞,f(x) → 0 −
As x →−∞, the graph is below y = x
As x →∞,f(x) → 0 + As x →∞, the graph is above y = x
19 This is hard to see on the calculator, but trust me, the graph is below the x-axis to the left of x = −7.