Rational Functions: Get Stronger Key — Precalculus Practice Page Answer Keys

Rational Functions: Get Stronger Key

Rational Functions

Solutions to Odd-Numbered 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.

7. [latex]\text{All reals }x\ne -1, 1[/latex]

9. [latex]\text{All reals }x\ne -1, -2, 1, 2[/latex]

11. V.A. at [latex]x=-\frac{2}{5}[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x\ne -\frac{2}{5}[/latex]

13. V.A. at [latex]x=4, -9[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x\ne 4, -9[/latex]

15. V.A. at [latex]x=0, 4, -4[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x\ne 0,4, -4[/latex]

17. V.A. at [latex]x=-5[/latex]; H.A. at [latex]y=0[/latex]; Domain is all reals [latex]x\ne 5,-5[/latex]

19. V.A. at [latex]x=\frac{1}{3}[/latex]; H.A. at [latex]y=-\frac{2}{3}[/latex]; Domain is all reals [latex]x\ne \frac{1}{3}[/latex].

21. none

23. [latex]x\text{-intercepts none, }y\text{-intercept }\left(0,\frac{1}{4}\right)[/latex]

25. Local behavior: [latex]x\to -{\frac{1}{2}}^{+},f\left(x\right)\to -\infty ,x\to -{\frac{1}{2}}^{-},f\left(x\right)\to \infty[/latex]

End behavior: [latex]x\to \pm \infty ,f\left(x\right)\to \frac{1}{2}[/latex]

27. Local behavior: [latex]x\to {6}^{+},f\left(x\right)\to -\infty ,x\to {6}^{-},f\left(x\right)\to \infty[/latex], End behavior: [latex]x\to \pm \infty ,f\left(x\right)\to -2[/latex]

29. Local behavior: [latex]x\to -{\frac{1}{3}}^{+},f\left(x\right)\to \infty ,x\to -{\frac{1}{3}}^{-}[/latex], [latex]f\left(x\right)\to -\infty ,x\to {\frac{5}{2}}^{-},f\left(x\right)\to \infty ,x\to {\frac{5}{2}}^{+}[/latex] , [latex]f\left(x\right)\to -\infty[/latex]

End behavior: [latex]x\to \pm \infty\\[/latex], [latex]f\left(x\right)\to \frac{1}{3}[/latex]

31. [latex]y=2x+4[/latex]

33. [latex]y=2x[/latex]

35. [latex]V.A.\text{ }x=0,H.A.\text{ }y=2[/latex]
Graph of a rational function.

37. [latex]V.A.\text{ }x=2,\text{ }H.A.\text{ }y=0[/latex]
Graph of a rational function.

39. [latex]V.A.\text{ }x=-4,\text{ }H.A.\text{ }y=2;\left(\frac{3}{2},0\right);\left(0,-\frac{3}{4}\right)[/latex]
Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.

41. [latex]V.A.\text{ }x=2,\text{ }H.A.\text{ }y=0,\text{ }\left(0,1\right)[/latex]
Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.

43. [latex]V.A.\text{ }x=-4,\text{ }x=\frac{4}{3},\text{ }H.A.\text{ }y=1;\left(5,0\right);\left(-\frac{1}{3},0\right);\left(0,\frac{5}{16}\right)[/latex]

45. [latex]V.A.\text{ }x=-1,\text{ }H.A.\text{ }y=1;\left(-3,0\right);\left(0,3\right)[/latex]
Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.

47. [latex]V.A.\text{ }x=4,\text{ }S.A.\text{ }y=2x+9;\left(-1,0\right);\left(\frac{1}{2},0\right);\left(0,\frac{1}{4}\right)[/latex]
Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.

49. [latex]V.A.\text{ }x=-2,\text{ }x=4,\text{ }H.A.\text{ }y=1,\left(1,0\right);\left(5,0\right);\left(-3,0\right);\left(0,-\frac{15}{16}\right)[/latex]
Graph of w(x)=(x-1)(x+3)(x-5)/(x+2)^2(x-4) with its vertical asymptotes at x=-2 and x=4 and horizontal asymptote at y=1.

51. [latex]y=50\frac{{x}^{2}-x - 2}{{x}^{2}-25}[/latex]

53. [latex]y=7\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20}[/latex]

55. [latex]y=\frac{1}{2}\frac{{x}^{2}-4x+4}{x+1}[/latex]

57. [latex]y=4\frac{x - 3}{{x}^{2}-x - 12}[/latex]

59. [latex]y=-9\frac{x - 2}{{x}^{2}-9}[/latex]

61. [latex]y=\frac{1}{3}\frac{{x}^{2}+x - 6}{x - 1}[/latex]

63. [latex]y=-6\frac{{\left(x - 1\right)}^{2}}{\left(x+3\right){\left(x - 2\right)}^{2}}[/latex]

65.

x 2.01 2.001 2.0001 1.99 1.999
y 100 1,000 10,000 –100 –1,000
x 10 100 1,000 10,000 100,000
y .125 .0102 .001 .0001 .00001

Vertical asymptote [latex]x=2[/latex], Horizontal asymptote [latex]y=0[/latex]

67.

x –4.1 –4.01 –4.001 –3.99 –3.999
y 82 802 8,002 –798 –7998
x 10 100 1,000 10,000 100,000
y 1.4286 1.9331 1.992 1.9992 1.999992

Vertical asymptote [latex]x=-4[/latex], Horizontal asymptote [latex]y=2[/latex]

69.

x –.9 –.99 –.999 –1.1 –1.01
y 81 9,801 998,001 121 10,201
x 10 100 1,000 10,000 100,000
y .82645 .9803 .998 .9998

Vertical asymptote [latex]x=-1[/latex], Horizontal asymptote [latex]y=1[/latex]

71. [latex]\left(\frac{3}{2},\infty \right)[/latex]
Graph of f(x)=4/(2x-3).

73. [latex]\left(-2,1\right)\cup \left(4,\infty \right)[/latex]
Graph of f(x)=(x+2)/(x-1)(x-4).

75. [latex]\left(2,4\right)[/latex]

77. [latex]\left(2,5\right)[/latex]

79. [latex]\left(-1,\text{1}\right)[/latex]

81. [latex]C\left(t\right)=\frac{8+2t}{300+20t}[/latex]

83. After about 6.12 hours.

85. [latex]A\left(x\right)=50{x}^{2}+\frac{800}{x}[/latex]. 2 by 2 by 5 feet.

87. [latex]A\left(x\right)=\pi {x}^{2}+\frac{100}{x}[/latex]. Radius = 2.52 meters.

Modeling Using Variation

Solutions to Odd-Numbered Exercises

1. The graph will have the appearance of a power function.

3. No. Multiple variables may jointly vary.

5. [latex]y=5{x}^{2}[/latex]

7. [latex]y=10{x}^{3}[/latex]

9. [latex]y=6{x}^{4}[/latex]

11. [latex]y=\frac{18}{{x}^{2}}[/latex]

13. [latex]y=\frac{81}{{x}^{4}}[/latex]

15. [latex]y=\frac{20}{\sqrt[3]{x}}[/latex]

17. [latex]y=10xzw[/latex]

19. [latex]y=10x\sqrt{z}[/latex]

21. [latex]y=4\frac{xz}{w}[/latex]

23. [latex]y=40\frac{xz}{\sqrt{w}{t}^{2}}[/latex]

25. [latex]y=256[/latex]

27. [latex]y=6[/latex]

29. [latex]y=6[/latex]

31. [latex]y=27[/latex]

33. [latex]y=3[/latex]

35. [latex]y=18[/latex]

37. [latex]y=90[/latex]

39. [latex]y=\frac{81}{2}[/latex]

41. [latex]y=\frac{3}{4}{x}^{2}[/latex]
Graph of y=3/4(x^2).

43. [latex]y=\frac{1}{3}\sqrt{x}[/latex]
Graph of y=1/3sqrt(x).

45. [latex]y=\frac{4}{{x}^{2}}[/latex]
Graph of y=4/(x^2).

47. 1.89 years

49. 0.61 years

51. 3 seconds

53. 48 inches

55. 49.75 pounds

57. 33.33 amperes

59. 2.88 inches