{"id":292,"date":"2026-01-30T23:00:20","date_gmt":"2026-01-30T23:00:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/rational-functions-get-stronger-key-precalculus-practice-page-answer-keys\/"},"modified":"2026-01-30T23:02:51","modified_gmt":"2026-01-30T23:02:51","slug":"rational-functions-get-stronger-key-precalculus-practice-page-answer-keys","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/rational-functions-get-stronger-key-precalculus-practice-page-answer-keys\/","title":{"raw":"Rational Functions: Get Stronger Key -- Precalculus Practice Page Answer Keys","rendered":"Rational Functions: Get Stronger Key — Precalculus Practice Page Answer Keys"},"content":{"raw":"\n\n\t\t
\n\t\t\t
\n\t
\n\t\t

<\/p>\n\t\t

Rational Functions: Get Stronger Key<\/h1>\n\t\t\t\t\t\t\t\t<\/div>\n\t
\n\t\t\t\t

Rational Functions<\/h2>

Solutions to Odd-Numbered Exercises<\/h2>

1. The rational function will be represented by a quotient of polynomial functions.<\/p>

3. The numerator and denominator must have a common factor.<\/p>

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.<\/p>

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

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

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]<\/p>

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]<\/p>

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]<\/p>

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]<\/p>

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].<\/p>

21. none<\/p>

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

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]<\/p>

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

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]<\/p>

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]<\/p>

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

31. [latex]y=2x+4[\/latex]<\/p>

33. [latex]y=2x[\/latex]<\/p>

35. [latex]V.A.\\text{ }x=0,H.A.\\text{ }y=2[\/latex]
\"Graph<\/p>

37. [latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0[\/latex]
\"Graph<\/p>

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<\/p>

41. [latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0,\\text{ }\\left(0,1\\right)[\/latex]
\"Graph<\/p>

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]<\/p>

45. [latex]V.A.\\text{ }x=-1,\\text{ }H.A.\\text{ }y=1;\\left(-3,0\\right);\\left(0,3\\right)[\/latex]
\"Graph<\/p>

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<\/p>

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<\/p>

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

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

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

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

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

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

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

65.<\/p>
x<\/em><\/td> 2.01<\/td> 2.001<\/td> 2.0001<\/td> 1.99<\/td> 1.999<\/td> <\/tr>
y<\/em><\/td> 100<\/td> 1,000<\/td> 10,000<\/td> \u2013100<\/td> \u20131,000<\/td> <\/tr> <\/tbody> <\/table>
x<\/em><\/td> 10<\/td> 100<\/td> 1,000<\/td> 10,000<\/td> 100,000<\/td> <\/tr> <\/tbody>
y<\/em><\/td> .125<\/td> .0102<\/td> .001<\/td> .0001<\/td> .00001<\/td> <\/tr> <\/tbody> <\/table>

Vertical asymptote [latex]x=2[\/latex], Horizontal asymptote [latex]y=0[\/latex]<\/p>

67.<\/p>
x<\/em><\/td> \u20134.1<\/td> \u20134.01<\/td> \u20134.001<\/td> \u20133.99<\/td> \u20133.999<\/td> <\/tr>
y<\/em><\/td> 82<\/td> 802<\/td> 8,002<\/td> \u2013798<\/td> \u20137998<\/td> <\/tr> <\/tbody> <\/table>
x<\/em><\/td> 10<\/td> 100<\/td> 1,000<\/td> 10,000<\/td> 100,000<\/td> <\/tr>
y<\/em><\/td> 1.4286<\/td> 1.9331<\/td> 1.992<\/td> 1.9992<\/td> 1.999992<\/td> <\/tr> <\/tbody> <\/table>

Vertical asymptote [latex]x=-4[\/latex], Horizontal asymptote [latex]y=2[\/latex]<\/p>

69.<\/p>
x<\/em><\/td> \u2013.9<\/td> \u2013.99<\/td> \u2013.999<\/td> \u20131.1<\/td> \u20131.01<\/td> <\/tr>
y<\/em><\/td> 81<\/td> 9,801<\/td> 998,001<\/td> 121<\/td> 10,201<\/td> <\/tr> <\/tbody> <\/table>
x<\/em><\/td> 10<\/td> 100<\/td> 1,000<\/td> 10,000<\/td> 100,000<\/td> <\/tr>
y<\/i><\/td> .82645<\/td> .9803<\/td> .998<\/td> .9998<\/td> <\/td> <\/tr> <\/tbody> <\/table>

Vertical asymptote [latex]x=-1[\/latex], Horizontal asymptote [latex]y=1[\/latex]<\/p>

71. [latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]
\"Graph<\/p>

73. [latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]
\"Graph<\/p>

75. [latex]\\left(2,4\\right)[\/latex]<\/p>

77. [latex]\\left(2,5\\right)[\/latex]<\/p>

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

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

83. After about 6.12 hours.<\/p>

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

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

Modeling Using Variation<\/h2>

Solutions to Odd-Numbered Exercises<\/h2>

1. The graph will have the appearance of a power function.<\/p>

3. No. Multiple variables may jointly vary.<\/p>

5. [latex]y=5{x}^{2}[\/latex]<\/p>

7. [latex]y=10{x}^{3}[\/latex]<\/p>

9. [latex]y=6{x}^{4}[\/latex]<\/p>

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

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

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

17. [latex]y=10xzw[\/latex]<\/p>

19. [latex]y=10x\\sqrt{z}[\/latex]<\/p>

21. [latex]y=4\\frac{xz}{w}[\/latex]<\/p>

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

25. [latex]y=256[\/latex]<\/p>

27. [latex]y=6[\/latex]<\/p>

29. [latex]y=6[\/latex]<\/p>

31. [latex]y=27[\/latex]<\/p>

33. [latex]y=3[\/latex]<\/p>

35. [latex]y=18[\/latex]<\/p>

37. [latex]y=90[\/latex]<\/p>

39. [latex]y=\\frac{81}{2}[\/latex]<\/p>

41. [latex]y=\\frac{3}{4}{x}^{2}[\/latex]
\"Graph<\/p>

43. [latex]y=\\frac{1}{3}\\sqrt{x}[\/latex]
\"Graph<\/p>

45. [latex]y=\\frac{4}{{x}^{2}}[\/latex]
\"Graph<\/p>

47. 1.89 years<\/p>

49. 0.61 years<\/p>

51. 3 seconds<\/p>

53. 48 inches<\/p>

55. 49.75 pounds<\/p>

57. 33.33 amperes<\/p>

59. 2.88 inches<\/p> \n\t<\/div>\n\t\t\t\n\t\t\t<\/div>\n\n\t\t<\/div>\n\t\n","rendered":"

\n
\n
\n

\n

Rational Functions: Get Stronger Key<\/h1>\n<\/p><\/div>\n
\n

Rational Functions<\/h2>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1. The rational function will be represented by a quotient of polynomial functions.<\/p>\n

3. The numerator and denominator must have a common factor.<\/p>\n

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.<\/p>\n

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

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

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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].<\/p>\n

21. none<\/p>\n

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

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]<\/p>\n

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

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]<\/p>\n

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]<\/p>\n

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

31. [latex]y=2x+4[\/latex]<\/p>\n

33. [latex]y=2x[\/latex]<\/p>\n

35. [latex]V.A.\\text{ }x=0,H.A.\\text{ }y=2[\/latex]
\"Graph<\/p>\n

37. [latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0[\/latex]
\"Graph<\/p>\n

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<\/p>\n

41. [latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0,\\text{ }\\left(0,1\\right)[\/latex]
\"Graph<\/p>\n

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]<\/p>\n

45. [latex]V.A.\\text{ }x=-1,\\text{ }H.A.\\text{ }y=1;\\left(-3,0\\right);\\left(0,3\\right)[\/latex]
\"Graph<\/p>\n

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<\/p>\n

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<\/p>\n

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

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

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

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

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

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

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

65.<\/p>\n\n\n\n\n
x<\/em><\/td>\n2.01<\/td>\n2.001<\/td>\n2.0001<\/td>\n1.99<\/td>\n1.999<\/td>\n<\/tr>\n
y<\/em><\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n\u2013100<\/td>\n\u20131,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
x<\/em><\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n<\/tbody>\n
y<\/em><\/td>\n.125<\/td>\n.0102<\/td>\n.001<\/td>\n.0001<\/td>\n.00001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Vertical asymptote [latex]x=2[\/latex], Horizontal asymptote [latex]y=0[\/latex]<\/p>\n

67.<\/p>\n\n\n\n\n
x<\/em><\/td>\n\u20134.1<\/td>\n\u20134.01<\/td>\n\u20134.001<\/td>\n\u20133.99<\/td>\n\u20133.999<\/td>\n<\/tr>\n
y<\/em><\/td>\n82<\/td>\n802<\/td>\n8,002<\/td>\n\u2013798<\/td>\n\u20137998<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
x<\/em><\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n
y<\/em><\/td>\n1.4286<\/td>\n1.9331<\/td>\n1.992<\/td>\n1.9992<\/td>\n1.999992<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Vertical asymptote [latex]x=-4[\/latex], Horizontal asymptote [latex]y=2[\/latex]<\/p>\n

69.<\/p>\n\n\n\n\n
x<\/em><\/td>\n\u2013.9<\/td>\n\u2013.99<\/td>\n\u2013.999<\/td>\n\u20131.1<\/td>\n\u20131.01<\/td>\n<\/tr>\n
y<\/em><\/td>\n81<\/td>\n9,801<\/td>\n998,001<\/td>\n121<\/td>\n10,201<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
x<\/em><\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n
y<\/i><\/td>\n.82645<\/td>\n.9803<\/td>\n.998<\/td>\n.9998<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Vertical asymptote [latex]x=-1[\/latex], Horizontal asymptote [latex]y=1[\/latex]<\/p>\n

71. [latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]
\"Graph<\/p>\n

73. [latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]
\"Graph<\/p>\n

75. [latex]\\left(2,4\\right)[\/latex]<\/p>\n

77. [latex]\\left(2,5\\right)[\/latex]<\/p>\n

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

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

83. After about 6.12 hours.<\/p>\n

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

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

Modeling Using Variation<\/h2>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1. The graph will have the appearance of a power function.<\/p>\n

3. No. Multiple variables may jointly vary.<\/p>\n

5. [latex]y=5{x}^{2}[\/latex]<\/p>\n

7. [latex]y=10{x}^{3}[\/latex]<\/p>\n

9. [latex]y=6{x}^{4}[\/latex]<\/p>\n

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

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

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

17. [latex]y=10xzw[\/latex]<\/p>\n

19. [latex]y=10x\\sqrt{z}[\/latex]<\/p>\n

21. [latex]y=4\\frac{xz}{w}[\/latex]<\/p>\n

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

25. [latex]y=256[\/latex]<\/p>\n

27. [latex]y=6[\/latex]<\/p>\n

29. [latex]y=6[\/latex]<\/p>\n

31. [latex]y=27[\/latex]<\/p>\n

33. [latex]y=3[\/latex]<\/p>\n

35. [latex]y=18[\/latex]<\/p>\n

37. [latex]y=90[\/latex]<\/p>\n

39. [latex]y=\\frac{81}{2}[\/latex]<\/p>\n

41. [latex]y=\\frac{3}{4}{x}^{2}[\/latex]
\"Graph<\/p>\n

43. [latex]y=\\frac{1}{3}\\sqrt{x}[\/latex]
\"Graph<\/p>\n

45. [latex]y=\\frac{4}{{x}^{2}}[\/latex]
\"Graph<\/p>\n

47. 1.89 years<\/p>\n

49. 0.61 years<\/p>\n

51. 3 seconds<\/p>\n

53. 48 inches<\/p>\n

55. 49.75 pounds<\/p>\n

57. 33.33 amperes<\/p>\n

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