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3.9e: Exercises - Rational Functions

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A: Conceptual Questions 

Exercise 3.9e.A

1. What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

2. What is the fundamental difference in the graphs of polynomial functions and rational functions?

3. If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

4. Can a graph of a rational function have no vertical asymptote? If so, how?

5. Can a graph of a rational function have no x-intercepts? If so, how?

Answers to odd exercises.

1. The rational function will be represented by a quotient of polynomial functions.
3. The numerator and denominator must have a common factor.
5. Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator. 

B: Find Domain of Rational Functions

Exercise 3.9e.B

 Find the domain of the rational functions.

7. f(x)=x+1x21 8. f(x)=x1x+2 9. f(x)=x2+4x3x45x2+4 10. f(x)=x2+4x22x8
Answers to odd exercises.

7. All reals except that x1,1
9. All reals except that x1,2,1,2 

C: Find domain, vertical asymptotes, holes, and horizontal asymptotes

Exercise 3.9e.C

For each function, find (a) the domain, vertical asymptotes and holes, and (b) the horizontal asymptote and points of intersection of the HA with the function.  

  1. f(x)=2x23x2+1
  2. f(x)=4x1
  3. f(x)=2x25x23x2
  1. f(x)=x+2x29
  2. f(x)=xx2+5x36
  3. f(x)=3+xx327
  1. f(x)=3x4x316x
  2. f(x)=x3xx3+9x2+14x
  3. f(x)=x+5x225
  1. f(x)=2x4x6
  2. f(x)=42x3x1

Identify the removable discontinuity ("hole").

  1. f(x)=x24x2
  2. f(x)=x3+1x+1
  1. f(x)=x2+x6x2
  1. f(x)=2x2+5x3x+3
  1. f(x)=x3+x2x+1
Answers to odd exercises.

11. Domain is all reals;  no VA, No holes;      H.A. at y=23, No intersection;
13. Domain is all reals x1,25, V.A. at x=25; Hole at (1,27);      H.A. at y=0; No intersection;
15. Domain is all reals x4,9, V.A. at x=4,9, No holes;      H.A. at y=0; Intersection at (0,0);
17. Domain is all reals x0,4,4, V.A. at x=0,4,4, No holes;      H.A. at y=0; Intersection at (43,0);
19. Domain is all reals x5,5, V.A. at x=5, Hole at (5,110);      H.A. at y=0; No intersection;
21. Domain is all reals x13, V.A. at x=13, No holes;      H.A. at y=23; No intersection;
25. (2,4)     27. (2,5)     29. (1,1)

D: Describe local and end behaviour

Exercise 3.9e.D

Describe the local and end behavior of the functions.

  1. f(x)=x2x+1
  2. f(x)=2xx6
  1. f(x)=2xx6
  2. f(x)=x24x+3x24x5
  1. f(x)=2x2326x2+13x5 
Answers to odd exercises.

31. Local behavior: x12+,f(x),x12,f(x)
      End behavior: x±,f(x)12

33. Local behavior: x6+,f(x),x6,f(x)
      End behavior: x±,f(x)2

35. Local behavior: x52,f(x),x52+,f(x)
      x13,f(x),x13+,f(x),        End behavior: x±,f(x)13

E: Find Slant Asymptote and its intersection with the function

Exercise 3.9e.E

 For each function, (a) Find the slant asymptote of the functions, (b) Determine any points of intersection between the function and this asymptote.

  1. f(x)=24x2+6x2x+1
  2. f(x)=4x2102x4
  1. f(x)=81x2183x2
  2. f(x)=6x35x3x2+4

40.   f(x)=x2+5x+4x1
40.1 f(x)=x33x2x+3x22x8

Answers to odd exercises.

37. y=2x+4 No point of intersection    39. y=2x  Intersection at (0,0)

F: Find Intercepts

Exercise 3.9e.F

Find the x- and y-intercepts for the functions. 

43. f(x)=xx2x

44. f(x)=x+5x2+4

45.  f(x)=x2+x+6x210x+24

46.  f(x)=x2+8x+7x2+11x+30

48.  f(x)=942x23x212

Answers to odd exercises.

43. none     45. x -intercepts none, y -intercept (0,14) 

G: Identify attributes and sketch

Exercise 3.9e.G

Find the (a) vertical asymptotes, (b) coordinates of any holes, (c) end behaviour asymptote (horizontal or slant), (d) coordinates of any points of intersection of the function with the end behaviour asymptote, (e) coordinates of any x-intercepts, (f) coordinates of the y-intercept. Use that information to sketch a graph.

  1. f(x)=42x3
  2. f(x)=2x+1
  3. p(x)=2x3x+4
  4. q(x)=x53x1
  1. s(x)=4(x2)2
  2. r(x)=5(x+1)2
  3. f(x)=x+2(x1)(x4)
  4. f(x)=2(x1)(x+2)
  1. f(x)=(x4)(x+3)2(x+1)(x4)(x1)2
  2. f(x)=(x2)(x+4)(x2)(x3)(x+1)2
  3. f(x)=(x3)(x+6)(x+2)(x+4)(x1)(x+6)
  4. f(x)=(x+1)(x3)(1)(x+2)(x+1)(x4)(x+3)
  1. f(x)=3x214x53x2+8x16
  2. g(x)=2x2+7x153x214+15
  3. a(x)=x2+2x3x21
  1. b(x)=x2x6x24
  2. h(x)=2x2+x1x4
  3. k(x)=2x23x20x5
  1. w(x)=(x1)(x+3)(x5)(x+2)2(x4)
  2. z(x)=(x+2)2(x5)(x3)(x+1)(x+4)
  1. f(x)=x2+2x
  2. f(x)=x2+x+1x+1
  3. f(x)=x23x+2x3
  4. f(x)=x3+4x2x4x2+2x8
  1. f(x)=x2+3x+2x1
  2. f(x)=x2+3x+2x+3
  3. f(x)=x2+4x+3x+4
  4. f(x)=x23xx4
  1. f(x)=x24x+3x2
  2. f(x)=x2x6x1
  3. f(x)=x2+2x8x+1
  4. f(x)=x3x220xx2+3x
  1. f(x)=4x12x33x24x+12
  2. f(x)=2x27x+10
  3. f(x)=x2+9x216
  4. f(x)=x24x2+2x15
  1. f(x)=x2xx2+7x+10
  2. f(x)=x2+4x+3x25x+4
  3. f(x)=x3x2x+12x211x+12
  4. f(x)=x32x2x+23x211x+6
  1. f(x)=x1x27x+10
  2. f(x)=x+2x2+x12
  3. f(x)=x21x3x24x+4
  4. f(x)=x2+7x+12x3+9x2+20x
  1. f(x)=x2+4x+3x2+6x+8
  2. f(x)=x22x3x2+2x
  3. f(x)=x3x2+x2
  4. f(x)=x26x+9x2+x6
  1. f(x)=x2+x12x24
  2. f(x)=x3x32x24x+8
  3. f(x)=(x3+3x2+9x)x2+x2
  4. f(x)=x4+x2x34x
Answers to odd exercises #51-67. More graphs at the end of this section

51. (32,)

CNX_Precalc_Figure_03_07_226.jpg

53. V.A. x=4, H.A. y=2; (32,0); (0,34)

CNX_Precalc_Figure_03_07_205.jpg

55. V.A. x=2, H.A. y=0, (0,1)

CNX_Precalc_Figure_03_07_207.jpg 

57. (2,1)(4,)

CNX_Precalc_Figure_03_07_228.jpg 

59. 
 3.9e #59.png

61.

 3.9e #61.png 

63. V.A. x=4, x=43, H.A. y=1; (5,0); (13,0); (0,516)

CNX_Precalc_Figure_03_07_209.jpg 

65. V.A. x=1, H.A. y=1; (3,0); (0,3)

CNX_Precalc_Figure_03_07_211.jpg 

67. V.A. x=4, S.A. y=2x+9; (1,0); (12,0); (0,14)

CNX_Precalc_Figure_03_07_213.jpg 

H: Construct an Equation from a Description

Exercise 3.9e.H

 Write an equation for a rational function with the given characteristics.

  1. Vertical asymptotes at x=5 and x=5, x-intercepts at (2,0) and (1,0)y -intercept at (0,4)
  2. Vertical asymptotes at x=4 and x=1,x -intercepts at (1,0) and (5,0),y -intercept at (0,7)
  3. Vertical asymptotes at x=4 and x=5,x -intercepts at (4,0) and (6,0), Horizontal asymptote at y=7
  4. Vertical asymptotes at x=3 and x=6,x -intercepts at (2,0) and (1,0), Horizontal asymptote at y=2
  5. Vertical asymptote at x=1, Double zero at x=2,y -intercept at (0,2)
  6. Vertical asymptote at x=3, Double zero at x=1,y -intercept at (0,4)
Answers to odd exercises.

105. y=50(x2)(x+1)(x5)(x+5)  107. y=7(x+6)(x4)(x+5)(x+4)  109. y=(x2)22(x+1) 

I: Construct an Equation from a Graph

Exercise 3.9e.I

Use the graphs to write an equation for the function.

111.

CNX_Precalc_Figure_03_07_217.jpg 

112.

CNX_Precalc_Figure_03_07_218.jpg 

113.

CNX_Precalc_Figure_03_07_219.jpg 

114.

CNX_Precalc_Figure_03_07_220.jpg 

115.

CNX_Precalc_Figure_03_07_221.jpg 

116.

CNX_Precalc_Figure_03_07_222.jpg 

117.

CNX_Precalc_Figure_03_07_223.jpg 

118.

CNX_Precalc_Figure_03_07_224.jpg 

 
Answers to odd exercises.

111. y=4(x3)(x4)(x+3)  113. y=27(x2)(x+3)(x3)2  

115. y=(x+3)(x2)3(x1)  117. y=6(x1)2(x+3)(x2)2 

J: Use tables to show behaviour

Exercise 3.9e.J

Make tables to show the behavior of the function near the vertical asymptote and horizontal asymptote

  1. f(x)=1x2
  1. f(x)=xx3
  1. f(x)=2xx+4
  1. f(x)=2x(x3)2
  1. f(x)=x2x2+2x+1
Answers to odd exercises.
121.
Vertical asymptote x=2
2.01 2.001 2.0001 1.99 1.999
100 1,000 10,000 –100 –1,000
Horizontal asymptote y=0
10 100 1,000 10,000 100,000
.125 .0102 .001 .0001 .00001

123. 
Vertical asymptote x=4
–4.1 –4.01 –4.001 –3.99 –3.999
82 802 8,002 –798 –7998
Horizontal asymptote y=2
10 100 1,000 10,000 100,000
1.4286 1.9331 1.992 1.9992 1.999992

125.
Vertical asymptote x=1
–.9 –.99 –.999 –1.1 –1.01
81 9,801 998,001 121 10,201
Horizontal asymptote y=1
10 100 1,000 10,000 100,000
.82645 .9803 .998 .9998 .99998
 

G: Additional Answers for section G (#71-101)

Exercise 3.9e.A

 Graphs of odd-numbered exercises #71-101 in section G

Additional answers to odd exercises.

69. V.A. x=2, x=4, H.A. y=1, (1,0); (5,0); (3,0); (0,1516)

CNX_Precalc_Figure_03_07_215.jpg 

71. 
3.9e #71.png
73. 
3.9e #73.png
75. 
3.9e #75.png 
77. 
3.9e #77.png
79. 
3.9e #79.png
81. 
3.9e #81.png 
83. 
3.9e #83.png
85. 
3.9e #85.png
87. 
3.9e #87.png 
89. 
3.9e #89.png
91. 
3.9e #91.png
93. 
3.9e #93.png
95. 
3.9e #95.png
97. 
3.9e #97.png
99. 
3.9e #99.png 
101. 
3.9e #101.png 
 

 


This page titled 3.9e: Exercises - Rational Functions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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