Section 5.4: Properties of rational functions
#1-16:
a) find the domain, express your answer using words
b) find the equation of the vertical asymptote (if any)
1) 𝑓(𝑥) =
2𝑥−6
𝑥+3
1a) domain:
x+3=0
x = -3
Answer: domain all real numbers except (-3)
1b) answer: vertical asymptote equation x = -3
3𝑥+12
3) 𝑓(𝑥) = 𝑥 2 −3𝑥−4
x2 – 3x – 4 = 0
(x+1)(x-4) = 0
x+1=0
x–4=0
x = -1
x=4
Answer: domain all real numbers except -1 and 4
3b) answer: vertical asymptote x = -1 and x = 4
5) 𝑓(𝑥) =
3𝑥 2 −10𝑥−8
𝑥 2 −4
x2 – 4 = 0
(x+2)(x-2)=0
x+2=0
x = -2
x-2=0
x=2
Answer: domain all real numbers except 2, -2
5b) answer: vertical asymptote x = 2 and x = -2
7) 𝑓(𝑥) =
𝑥 2 −16
𝑥 3 −𝑥 2
x3 – x2 = 0
x2(x-1) = 0
x2 = 0
x–1=0
𝑥 = ±√0
x=0
x=1
Answer: domain all real numbers except 0 and 1
7b) answer: vertical asymptote x = 0 and x = 1
9) 𝑓(𝑥) =
3
𝑥−6
x- 6 = 0
x=6
Answer: domain all real numbers except 6
9b) answer: vertical asymptote x = 6
11) 𝑓(𝑥) =
2𝑥
𝑥 2 −9
x2 – 9 = 0
(x+3)(x-3) = 0
x+3 = 0
x = -3
x–3=0
x=3
Answer: domain all real numbers except 3 and -3
11b) answer: vertical asymptote x = 3 and x = -3
13) 𝑓(𝑥) =
𝑥+2
𝑥 2 +16
x2 + 16 = 0 (this is prime so I will solve using square roots)
x2 = -16
𝑥 = ±√−16
x = ±4𝑖 (since all answers have i there aren’t any numbers to exclude from the domain)
Answer: domain all real numbers
13b) answer: vertical asymptote none
5𝑥
15) 𝑓(𝑥) = 𝑥 2 +9
x2 + 9 = 0 (this is prime so I will solve using square roots)
x2 = -9
𝑥 = ±√−9
x = ±3𝑖 (since all answers have i there aren’t any numbers to exclude from the domain)
Answer: domain all real numbers
15b) answer: vertical asymptote none
#1 7– 32
a) find the x-intercept
b) find the y-intercept
17) 𝑓(𝑥) =
2𝑥−6
𝑥+3
x-intercept
2x – 6 = 0
2x = 6
x=3
y-intercept
𝑓(0) =
2∗0−6
0+3
=
−6
3
= −2
Answer: x-intercept (3,0)
y-intercept (0,-2)
19) 𝑓(𝑥) =
3𝑥+12
𝑥 2 −3𝑥−4
x-intercept
3x + 12 = 0
3x = -12
x = -4
y-intercept
3∗0+12
12
𝑓(0) = 02 −3∗0−4 = −4 = −3
Answer: x-intercept (-4,0)
21) 𝑓(𝑥) =
y-intercept (0,-3)
3𝑥 2 −10𝑥−8
𝑥 2 −4
x-intercept
3x2 – 10x – 8 = 0
(3x+2)(x-4) = 0
3x+2 = 0
x-4=0
x = -2/3
x=4
y-intercept
𝑓(0) =
3∗02 −10∗0−8
02 −4
−8
= −4 = 2
Answer: x-intercept (-2/3,0)
23) 𝑓(𝑥) =
(4,0)
y-intercept (0,2)
𝑥 2 −16
𝑥 3 −𝑥 2
x-intercept
x2 – 16 = 0
(x+4)(x-4) = 0
x+4=0
x–4=0
x = -4
x=4
y-intercept
02 −16
𝑓(0) = 03 −02 =
−16
0
= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Answer: x-intercept (-4,0)
(4,0)
y-intercept none
25) 𝑓(𝑥) =
3
𝑥−6
x-intercept
3 = 0 (no x so there is no solution and no x-intercept
y-intercept
3
3
𝑓(0) = 0−6 = −6 =
−1
2
Answer: x-intercept none
y-intercept (0, -1/2)
2𝑥
27) 𝑓(𝑥) = 𝑥 2 −9
x-intercept
2x = 0
x = 0/2
x=0
y-intercept
𝑓(0) =
2∗0
02 −9
=
0
−9
=0
Answer: x-intercept (0,0)
y-intercept (0,0)
𝑥+2
29) 𝑓(𝑥) = 𝑥 2 +16
x-intercept
x+ 2 = 0
x = -2
y-intercept
𝑓(0) =
0+2
02 +16
=
2
16
=
1
8
Answer: x-intercept (-2,0)
y-intercept (0, 1/8)
31) 𝑓(𝑥) =
5𝑥
𝑥 2 +9
x-intercept
5x = 0
x = 0/5
x=0
y-intercept
5∗0
0
𝑓(0) = 02 +9 = 9 = 0
Answer: x-intercept (0,0)
33) 𝑓(𝑥) =
y-intercept (0,0)
2𝑥−6
𝑥+3
2
1
𝑦= =2
Answer: y = 2
3𝑥+12
35) 𝑓(𝑥) = 𝑥 2 −3𝑥−4
𝑦 = 𝑎𝑛𝑠𝑤𝑒𝑟 𝑦 = 0 𝑠𝑖𝑛𝑐𝑒 𝑙𝑎𝑟𝑔𝑒𝑟 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑛 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
Answer: y = 0
37) 𝑓(𝑥) =
3𝑥 2 −10𝑥−8
𝑥 2 −4
3
𝑦=1=3
Answer: y = 3
39) 𝑓(𝑥) =
𝑥 2 −16
𝑥 2 −𝑥
1
𝑦=1=1
Answer: y = 1
41) 𝑓(𝑥) =
3
𝑥−6
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 𝑖𝑛 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑠𝑜 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 𝑦 = 0
Answer: y = 0
2𝑥
43) 𝑓(𝑥) = 𝑥 2 −9
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 𝑖𝑛 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑠𝑜 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 𝑦 = 0
Answer: y = 0
𝑥+2
45) 𝑓(𝑥) = 𝑥 2 +16
𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 𝑖𝑛 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑠𝑜 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 𝑦 = 0
Answer: y = 0
5𝑥 2
47) 𝑓(𝑥) = 𝑥 2 +9
5
1
𝑦= =5
Answer: 𝒚 = 𝟓
49) 𝑓(𝑥) =
𝑦=
𝑥2
2𝑥
=
𝑥
2
𝑥 2 +5𝑥+1
2𝑥+3
1
2
= 𝑥
𝟏
Answer: 𝒚 = 𝟐 𝒙
51) 𝑓(𝑥) =
𝑦=
12𝑥 2
4𝑥
12𝑥 2 +5𝑥+1
4𝑥−5
= 3𝑥
Answer: y = 3x
53) 𝑓(𝑥) =
𝑦=
2𝑥 2
4𝑥
2𝑥 2
4𝑥−1
𝑥
2
1
2
= = 𝑥
𝟏
Answer: 𝒚 = 𝟐 𝒙
55) 𝑓(𝑥) =
𝑦=
2𝑥 3
6𝑥 2
2𝑥 3 +3𝑥 2 −5
6𝑥 2 +6𝑥−1
𝑥
3
1
3
= = 𝑥
𝟏
Answer: 𝒚 = 𝟑 𝒙
57) 𝑓(𝑥) =
𝑥3
𝑥 2 +1
𝑥3
𝑦 = 𝑥2 = 𝑥
Answer: y = x