Section 5.5 solutions
1. f ( x) 
3x  6
x6
a) Domain
x- 6 = 0
x=6
Answer: domain all real numbers except x = 6
b) Vertical Asymptote (if any)
Answer: x = 6
c) Horizontal asymptote, or slant asymptote
𝑦=
3𝑥
𝑥
=3
Answer: y = 3
d) x- intercept(s) if any
3x+6 = 0
3x = -6
x = -2
Answer: (-2,0)
e) y-intercept(s) if any
𝑓(0) =
3∗0+6
0−6
6
= −6 = −1
Answer: (0,-1)
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min = -30 x-max = 30 y-min = -30 y-max = 30
3.
f ( x) 
4x 2  9
x2 1
a) Domain
x2 – 1 = 0
x2 = 1
𝑥 = ±√1 (I could have also solved by factoring)
Answer: domain all real numbers except 1, -1
b) Vertical Asymptote (if any)
Answer: x = 1 and x = -1
c) Horizontal asymptote, or slant asymptote
d) x- intercept(s) if any
4x2 – 9 = 0
(2x+3)(2x-3) = 0
2x+3=0
2x-3 = 0
x = -3/2
x = 3/2
Answer: (-3/2, 0) (3/2,0)
4𝑥 2
𝑥2
𝑦=
=4
Answer: y = 4
e) y-intercept(s) if any
𝑓(0) =
4∗02 −9
02 −1
−9
= −1 = 9
Answer: (9,0)
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min -10 x-max 10 y-min -10 y-max 20
5. f ( x ) 
3
5x
a) Domain
5x = 0
x = 0/5
x= 0
Answer: all real numbers except 0
b) Vertical Asymptote (if any)
Answer: x = 0 (this is the y-axis)
c) Horizontal asymptote, or slant asymptote
3
𝑦 = 5𝑥
Since x left in denominator x-axis will be horizontal
asymptote
Answer: y = 0
d) x- intercept(s) if any
3 = 0 (no x so no solution)
Answer: none
e) y-intercept(s) if any
𝑓(0) =
3
5∗0
3
0
= = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑, 𝑠𝑜 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 𝑛𝑜 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠
Answer: none
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min -5 x-max 5 y-min -5 y-max 5
7.
f ( x) 
x3
x 2  16
a) Domain
x2 – 16 = 0
(x+4)(x-4)=0
x = 4, x = -4
answer: all real numbers except 4, -4
b) Vertical Asymptote (if any)
Answer: x = 4 and x = -4
c) Horizontal asymptote, or slant asymptote
𝑥
1
𝑦= 2=
𝑥
𝑥
Answer: y = 0
d) x- intercept(s) if any
x+ 3 = 0
x = -3
answer: (-3,0)
e) y-intercept(s) if any
𝑓(0) =
0+3
02 −16
=
3
−16
=
−3
16
Answer: (0, -3/16)
f) Sketch a graph of the function (label all features found in parts b – e)
window x-min -10 x-max 10 y-min -10 y-max 10
3x 2  8 x  5
9. f ( x) 
x5
a) Domain
x- 5 = 0
x=5
Answer: all real numbers except 5
b) Vertical Asymptote (if any)
answer x = 5
c) Horizontal asymptote, or slant asymptote
d) x- intercept(s) if any
3x2 + 8x + 5 = 0
(3x + 5)(x+1) = 0
3x + 5 = 0
x+1=0
x = -5/3
x = -1
Answer: (-5/3,0) (-1,0)
3𝑥 2
𝑥
𝑦=
= 3𝑥
Answer: slant asymptote y = 3x
e) y-intercept(s) if any
𝑓(0) =
3∗02 +8∗0+5
0−5
5
= −5 = −1
Answer: (0,-1)
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min -50 x-max 50 y-min -200 y-max 200 (hard to see intercepts with this window)
11) 𝑓(𝑥) =
2𝑥 2
4𝑥−1
a) Domain
4x-1 = 0
x=¼
Answer: all real numbers except 1/4
b) Vertical Asymptote (if any)
answer: x = 1/4
c) Horizontal asymptote, or slant asymptote
d) x- intercept(s) if any
2x2 = 0 (divide by 2)
x2 = 0
𝑥 = ±√0
x=0
Answer: (0,0)
𝑦=
2𝑥 2
4𝑥
𝑥
2
= =
1
𝑥
2
𝟏
Answer: slant asymptote y = 𝟐 𝒙
e) y-intercept(s) if any
2∗02
0
𝑓(0) = 4∗0−1 = −1 = 0
Answer: (0,0)
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min -10 x-max 10 y-min -10 y-max 10
13) f ( x) 
3 x  12
2x  6
a) Domain
2x+6 = 0
2x = -6
x = -3
Answer: all real numbers except -3
b) Vertical Asymptote (if any)
Answer: x = -3
c) Horizontal asymptote, or slant asymptote
3𝑥
3
𝑦 = 2𝑥 = 2
Answer: y = 3/2
d) x- intercept(s) if any
3x -12 = 0
3x = 12
Answer: (4,0)
e) y-intercept(s) if any
𝑓(0) =
3∗0−12
2∗0+6
=
−12
6
= −2
Answer (0,-2)
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min -20 x-max 20 y-min -20 y-max 20
15) f ( x ) 
1
x
a) Domain
x=0
Answer: all real numbers except 0
b) Vertical Asymptote (if any)
answer: x = 0
c) Horizontal asymptote, or slant asymptote
fraction already in form to determine horizontal
asymptote
Answer: y = 0
d) x- intercept(s) if any
1 = 0 (no x so no solution)
Answer: none
e) y-intercept(s) if any
1
𝑓(0) = 0 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Answer: none
f) Sketch a graph of the function (label all features found in parts b – e)
window: x-min -10 x-max 10 y-min -10 y-max 10
17) f ( x ) 
x5
x2
a) Domain
x2 = 0
𝑥 = ±√0
x=0
Answer: all real numbers except 0
b) Vertical Asymptote (if any)
Answer: x = 0
c) Horizontal asymptote, or slant asymptote
𝑥
1
𝑦= 2=
𝑥
𝑥
Answer: y = 0
d) x- intercept(s) if any
x+5 = 0
x = -5
Answer (-5,0)
e) y-intercept(s) if any
𝑓(0) =
0+5
02
5
= 0 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Answer: none
f) Sketch a graph of the function (label all features found in parts b – e)
window used x-min -10 x-max 10 y-min -2 y-max 2 (I can’t make a window show the what
happens around the x-intercept and have it approach the vertical asymptote. My graph focuses
more on the x-intercept region of the graph.