{"id":2400,"date":"2019-05-14T13:51:19","date_gmt":"2019-05-14T13:51:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2400"},"modified":"2019-05-14T13:51:20","modified_gmt":"2019-05-14T13:51:20","slug":"4-3-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/4-3-section-exercises\/","title":{"raw":"4.3 Section Exercises","rendered":"4.3 Section Exercises"},"content":{"raw":"
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4.3 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137848972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137848972\"]\r\n

The rational function will be represented by a quotient of polynomial functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. What is the fundamental difference in the graphs of polynomial functions and rational functions?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n3. If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135518222\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135518222\"]\r\n

The numerator and denominator must have a common factor.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Can a graph of a rational function have no vertical asymptote? If so, how?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. Can a graph of a rational function have no x<\/em>-intercepts? If so, how?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135393372\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135393372\"]\r\n

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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n

For the following exercises, find the domain of the rational functions.<\/p>\r\n\r\n

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6. [latex]f\\left(x\\right)=\\frac{x-1}{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n7. [latex]f\\left(x\\right)=\\frac{x+1}{{x}^{2}-1}[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165134122925\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134122925\"]\r\n

[latex]\\text{All reals }x\\ne \u20131, 1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [latex]f\\left(x\\right)=\\frac{{x}^{2}+4}{{x}^{2}-2x-8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]f\\left(x\\right)=\\frac{{x}^{2}+4x-3}{{x}^{4}-5{x}^{2}+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137619914\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137619914\"]\r\n

[latex]\\text{All reals }x\\ne \u20131, \u20132, 1, 2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.<\/p>\r\n\r\n

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10. [latex]f\\left(x\\right)=\\frac{4}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]f\\left(x\\right)=\\frac{2}{5x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135181414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135181414\"]\r\n

V.A. at[latex]\\text{ }x=\u2013\\frac{2}{5};\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne \u2013\\frac{2}{5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}-9}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}+5x-36}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"38670\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"38670\"]\u00a0V.A. at[latex]\\text{ }x=4, \u20139;\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne 4, \u20139[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]f\\left(x\\right)=\\frac{3+x}{{x}^{3}-27}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]f\\left(x\\right)=\\frac{3x-4}{{x}^{3}-16x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135443834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135443834\"]\r\n

V.A. at[latex]\\text{ }x=0, 4, -4;\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;[\/latex]Domain is all reals[latex]\\text{ }x\\ne 0,4, \u20134[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]f\\left(x\\right)=\\frac{{x}^{2}-1}{{x}^{3}+9{x}^{2}+14x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]f\\left(x\\right)=\\frac{x+5}{{x}^{2}-25}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135192050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135192050\"]\r\n

V.A. at[latex]\\text{ }x=-5;\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne 5,-5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]f\\left(x\\right)=\\frac{x-4}{x-6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]f\\left(x\\right)=\\frac{4-2x}{3x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137737944\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137737944\"]\r\n

V.A. at[latex]\\text{ }x=\\frac{1}{3};\\text{ }[\/latex]H.A. at[latex]\\text{ }y=-\\frac{2}{3};\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne \\frac{1}{3}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the x<\/em>- and y<\/em>-intercepts for the functions.<\/p>\r\n\r\n

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20. [latex]f\\left(x\\right)=\\frac{x+5}{{x}^{2}+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}-x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137727815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137727815\"]\r\n

none<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]f\\left(x\\right)=\\frac{{x}^{2}+8x+7}{{x}^{2}+11x+30}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]f\\left(x\\right)=\\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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

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\r\n\r\n25. [latex]f\\left(x\\right)=\\frac{94-2{x}^{2}}{3{x}^{2}-12}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, describe the local and end behavior of the functions.<\/p>\r\n\r\n

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26. [latex]f\\left(x\\right)=\\frac{x}{2x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137527922\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137527922\"]\r\n

Local behavior:[latex]\\text{ }x\\to -{\\frac{1}{2}}^{+},f\\left(x\\right)\\to -\\infty ,x\\to -{\\frac{1}{2}}^{-},f\\left(x\\right)\\to \\infty \\text{ }[\/latex]<\/p>\r\n

End behavior:[latex]\\text{ }x\\to \u00b1\\infty ,f\\left(x\\right)\\to \\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]f\\left(x\\right)=\\frac{2x}{x-6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. [latex]f\\left(x\\right)=\\frac{-2x}{x-6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137793829\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137793829\"]\r\n

Local behavior:[latex]\\text{ }x\\to {6}^{+},f\\left(x\\right)\\to -\\infty ,x\\to {6}^{-},f\\left(x\\right)\\to \\infty ,\\text{ }[\/latex]End behavior:[latex]\\text{ }x\\to \u00b1\\infty ,f\\left(x\\right)\\to -2[\/latex]<\/p>\r\n29. [\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4x+3}{{x}^{2}-4x-5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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31. [latex]f\\left(x\\right)=\\frac{2{x}^{2}-32}{6{x}^{2}+13x-5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134073889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134073889\"]\r\n

Local behavior:[latex]\\text{ }x\\to -{\\frac{1}{3}}^{+},f\\left(x\\right)\\to \\infty ,x\\to -{\\frac{1}{3}}^{-},\\text{ }[\/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>\r\nEnd behavior: [latex]x\\to \u00b1\\infty ,[\/latex][latex]f\\left(x\\right)\\to \\frac{1}{3}[\/latex]\r\n

[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the slant asymptote of the functions.<\/p>\r\n\r\n

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32. [latex]f\\left(x\\right)=\\frac{24{x}^{2}+6x}{2x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n33. [latex]f\\left(x\\right)=\\frac{4{x}^{2}-10}{2x-4}[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137803153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137803153\"]\r\n

[latex]y=2x+4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]f\\left(x\\right)=\\frac{81{x}^{2}-18}{3x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [latex]f\\left(x\\right)=\\frac{6{x}^{3}-5x}{3{x}^{2}+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135152039\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135152039\"]\r\n

[latex]y=2x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. [latex]f\\left(x\\right)=\\frac{{x}^{2}+5x+4}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Graphical<\/h4>\r\n

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.<\/p>\r\n\r\n

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37. The reciprocal function shifted up two units.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137749319\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137749319\"]\r\n

[latex]V.A.\\text{ }x=0,H.A.\\text{ }y=2[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. The reciprocal function shifted down one unit and left three units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. The reciprocal squared function shifted to the right 2 units.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137673413\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137673413\"]\r\n

[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. The reciprocal squared function shifted down 2 units and right 1 unit.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.\r\n

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\r\n\r\n41. [latex]p\\left(x\\right)=\\frac{2x-3}{x+4}[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137832470\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832470\"]\r\n

[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]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [latex]q\\left(x\\right)=\\frac{x-5}{3x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n43. [latex]s\\left(x\\right)=\\frac{4}{{\\left(x-2\\right)}^{2}}[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135533080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135533080\"]\r\n

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

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44. [latex]r\\left(x\\right)=\\frac{5}{{\\left(x+1\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. [latex]f\\left(x\\right)=\\frac{3{x}^{2}-14x-5}{3{x}^{2}+8x-16}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135195965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135195965\"]\r\n

[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>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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46. [latex]g\\left(x\\right)=\\frac{2{x}^{2}+7x-15}{3{x}^{2}-14+15}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. [latex]a\\left(x\\right)=\\frac{{x}^{2}+2x-3}{{x}^{2}-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137635057\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137635057\"]\r\n

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

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48. [latex]b\\left(x\\right)=\\frac{{x}^{2}-x-6}{{x}^{2}-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [latex]h\\left(x\\right)=\\frac{2{x}^{2}+ x-1}{x-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135208397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135208397\"]\r\n

[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]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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50. [latex]k\\left(x\\right)=\\frac{2{x}^{2}-3x-20}{x-5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. [latex]w\\left(x\\right)=\\frac{\\left(x-1\\right)\\left(x+3\\right)\\left(x-5\\right)}{{\\left(x+2\\right)}^{2}\\left(x-4\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135264668\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135264668\"]\r\n

[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]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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52. [latex]z\\left(x\\right)=\\frac{{\\left(x+2\\right)}^{2}\\left(x-5\\right)}{\\left(x-3\\right)\\left(x+1\\right)\\left(x+4\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, write an equation for a rational function with the given characteristics.<\/p>\r\n\r\n

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53. Vertical asymptotes at[latex]\\text{ }x=5\\text{ }[\/latex]and[latex]\\text{ }x=-5,\\text{ }[\/latex]x<\/em>-intercepts at[latex]\\text{ }\\left(2,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(-1,0\\right),\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137766733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137766733\"]\r\n

[latex]y=50\\frac{{x}^{2}-x-2}{{x}^{2}-25}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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54. Vertical asymptotes at[latex]\\text{ }x=-4\\text{ }[\/latex]and[latex]\\text{ }x=-1,\\text{ }[\/latex]x-<\/em>intercepts at[latex]\\text{ }\\left(1,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(5,0\\right),\\text{ }[\/latex]y-<\/em>intercept at[latex]\\text{ }\\left(0,7\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55. Vertical asymptotes at[latex]\\text{ }x=-4\\text{ }[\/latex]and[latex]\\text{ }x=-5,\\text{ }[\/latex]x<\/em>-intercepts at[latex]\\text{ }\\left(4,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(-6,0\\right),\\text{ }[\/latex]Horizontal asymptote at[latex]\\text{ }y=7[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134328215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134328215\"]\r\n

[latex]y=7\\frac{{x}^{2}+2x-24}{{x}^{2}+9x+20}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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56. Vertical asymptotes at[latex]\\text{ }x=-3\\text{ }[\/latex]and[latex]\\text{ }x=6,\\text{ }[\/latex]x<\/em>-intercepts at[latex]\\text{ }\\left(-2,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(1,0\\right),\\text{ }[\/latex]Horizontal asymptote at[latex]\\text{ }y=-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. Vertical asymptote at[latex]\\text{ }x=-1,\\text{ }[\/latex]Double zero at[latex]\\text{ }x=2,\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137832215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832215\"]\r\n

[latex]y=\\frac{1}{2}\\frac{{x}^{2}-4x+4}{x+1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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58. Vertical asymptote at[latex]\\text{ }x=3,\\text{ }[\/latex]Double zero at[latex]\\text{ }x=1,\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the graphs to write an equation for the function.<\/p>\r\n\r\n

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59.\u00a0\"Graph<\/span><\/div>\r\n
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[reveal-answer q=\"856098\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"856098\"][latex]y=4\\frac{x-3}{{x}^{2}-x-12}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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60.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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61.\u00a0\"Graph<\/span><\/div>\r\n
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[reveal-answer q=\"870551\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"870551\"][latex]y=-9\\frac{x-2}{{x}^{2}-9}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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62.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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63.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137414994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137414994\"]\r\n

[latex]y=\\frac{1}{3}\\frac{{x}^{2}+x-6}{x-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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64.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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65.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137732796\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137732796\"]\r\n

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

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66.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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Numeric<\/h4>\r\n

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote<\/p>\r\n\r\n

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67. [latex]f\\left(x\\right)=\\frac{1}{x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"937443\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"937443\"]\r\n

\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/td>\r\n2.01<\/td>\r\n2.001<\/td>\r\n2.0001<\/td>\r\n1.99<\/td>\r\n1.999<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex]<\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n\u2013100<\/td>\r\n\u20131,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/th>\r\n10<\/th>\r\n100<\/th>\r\n1,000<\/th>\r\n10,000<\/th>\r\n100,000<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]y[\/latex]<\/td>\r\n.125<\/td>\r\n.0102<\/td>\r\n.001<\/td>\r\n.0001<\/td>\r\n.00001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Vertical asymptote[latex]\\text{ }x=2,\\text{ }[\/latex]Horizontal asymptote[latex]\\text{ }y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

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\r\n\r\n68. [latex]f\\left(x\\right)=\\frac{x}{x-3}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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69. [latex]f\\left(x\\right)=\\frac{2x}{x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135501151\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135501151\"]\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/td>\r\n\u20134.1<\/td>\r\n\u20134.01<\/td>\r\n\u20134.001<\/td>\r\n\u20133.99<\/td>\r\n\u20133.999<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex]<\/td>\r\n82<\/td>\r\n802<\/td>\r\n8,002<\/td>\r\n\u2013798<\/td>\r\n\u20137998<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/td>\r\n10<\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n100,000<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex]<\/td>\r\n1.4286<\/td>\r\n1.9331<\/td>\r\n1.992<\/td>\r\n1.9992<\/td>\r\n1.999992<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Vertical asymptote[latex]\\text{ }x=-4,\\text{ }[\/latex]Horizontal asymptote[latex]\\text{ }y=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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70. [latex]f\\left(x\\right)=\\frac{2x}{{\\left(x-3\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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71. [latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}+2x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137779967\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137779967\"]\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/td>\r\n\u2013.9<\/td>\r\n\u2013.99<\/td>\r\n\u2013.999<\/td>\r\n\u20131.1<\/td>\r\n\u20131.01<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex]<\/td>\r\n81<\/td>\r\n9,801<\/td>\r\n998,001<\/td>\r\n121<\/td>\r\n10,201<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
\u00a0<\/caption>\r\n
[latex]x[\/latex]<\/td>\r\n10<\/td>\r\n100<\/td>\r\n1,000<\/td>\r\n10,000<\/td>\r\n100,000<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex]<\/td>\r\n.82645<\/td>\r\n.9803<\/td>\r\n.998<\/td>\r\n.9998<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Vertical asymptote[latex]\\text{ }x=-1,\\text{ }[\/latex]Horizontal asymptote[latex]\\text{ }y=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Technology<\/h4>\r\n

For the following exercises, use a calculator to graph[latex]\\text{ }f\\left(x\\right).\\text{ }[\/latex]Use the graph to solve[latex]\\text{ }f\\left(x\\right)>0.[\/latex]<\/p>\r\n\r\n

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72. [latex]f\\left(x\\right)=\\frac{2}{x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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73. [latex]f\\left(x\\right)=\\frac{4}{2x-3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135551843\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135551843\"]\r\n

[latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]<\/p>\r\n\"Graph[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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74. [latex]f\\left(x\\right)=\\frac{2}{\\left(x-1\\right)\\left(x+2\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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75. [latex]f\\left(x\\right)=\\frac{x+2}{\\left(x-1\\right)\\left(x-4\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137742755\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137742755\"]\r\n

[latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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76. [latex]f\\left(x\\right)=\\frac{{\\left(x+3\\right)}^{2}}{{\\left(x-1\\right)}^{2}\\left(x+1\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Extensions<\/h4>\r\n

For the following exercises, identify the removable discontinuity.<\/p>\r\n\r\n

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77. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4}{x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135195746\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135195746\"]\r\n

[latex]\\left(2,4\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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78. [latex]f\\left(x\\right)=\\frac{{x}^{3}+1}{x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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79. [latex]f\\left(x\\right)=\\frac{{x}^{2}+x-6}{x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134261663\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134261663\"]\r\n

[latex]\\left(2,5\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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80. [latex]f\\left(x\\right)=\\frac{2{x}^{2}+5x-3}{x+3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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81. [latex]f\\left(x\\right)=\\frac{{x}^{3}+{x}^{2}}{x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137762314\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137762314\"]\r\n

[latex]\\left(\u20131,\\text{1}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Real-World Applications<\/h4>\r\n

For the following exercises, express a rational function that describes the situation.<\/p>\r\n\r\n

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82. A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after[latex]\\text{ }t\\text{ }[\/latex]minutes.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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83. A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after[latex]\\text{ }t\\text{ }[\/latex]minutes.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135581177\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135581177\"]\r\n

[latex]C\\left(t\\right)=\\frac{8+2t}{300+20t}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the given rational function to answer the question.<\/p>\r\n\r\n

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84. The concentration[latex]\\text{ }C\\text{ }[\/latex]of a drug in a patient\u2019s bloodstream[latex]\\text{ }t\\text{ }[\/latex]hours after injection in given by[latex]\\text{ }C\\left(t\\right)=\\frac{2t}{3+{t}^{2}}.\\text{ }[\/latex]What happens to the concentration of the drug as[latex]\\text{ }t\\text{ }[\/latex]increases?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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85. The concentration[latex]\\text{ }C\\text{ }[\/latex]of a drug in a patient\u2019s bloodstream[latex]\\text{ }t\\text{ }[\/latex]<\/sub>hours after injection is given by[latex]\\text{ }C\\left(t\\right)=\\frac{100t}{2{t}^{2}+75}.\\text{ }[\/latex]Use a calculator to approximate the time when the concentration is highest.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135368500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135368500\"]\r\n

After about 6.12 hours.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.<\/p>\r\n\r\n

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86. An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let[latex]\\text{ }x\\text{ }[\/latex]= length of the side of the base.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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87. A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents\/ square foot. The material for the sides costs 10 cents\/square foot. The material for the top costs 20 cents\/square foot. Determine the dimensions that will yield minimum cost. Let[latex]\\text{ }x\\text{ }[\/latex]= length of the side of the base.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137940592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137940592\"]\r\n

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

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88. A right circular cylinder has volume of 100 cubic inches. Find the radius and height that will yield minimum surface area. Let[latex]\\text{ }x\\text{ }[\/latex]= radius.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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89. A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let[latex]\\text{ }x\\text{ }[\/latex]= radius.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137436431\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137436431\"]\r\n

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

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90. A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents\/square inch to construct the top and bottom and 1 cent\/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let[latex]\\text{ }x\\text{ }[\/latex]= radius.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

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4.3 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The rational function will be represented by a quotient of polynomial functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2. What is the fundamental difference in the graphs of polynomial functions and rational functions?<\/p>\n<\/div>\n<\/div>\n

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3. If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The numerator and denominator must have a common factor.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4. Can a graph of a rational function have no vertical asymptote? If so, how?<\/p>\n<\/div>\n<\/div>\n

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5. Can a graph of a rational function have no x<\/em>-intercepts? If so, how?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, find the domain of the rational functions.<\/p>\n

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

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7. [latex]f\\left(x\\right)=\\frac{x+1}{{x}^{2}-1}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{All reals }x\\ne \u20131, 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8. [latex]f\\left(x\\right)=\\frac{{x}^{2}+4}{{x}^{2}-2x-8}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]f\\left(x\\right)=\\frac{{x}^{2}+4x-3}{{x}^{4}-5{x}^{2}+4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{All reals }x\\ne \u20131, \u20132, 1, 2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.<\/p>\n

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10. [latex]f\\left(x\\right)=\\frac{4}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [latex]f\\left(x\\right)=\\frac{2}{5x+2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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V.A. at[latex]\\text{ }x=\u2013\\frac{2}{5};\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne \u2013\\frac{2}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}-9}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}+5x-36}[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
\u00a0V.A. at[latex]\\text{ }x=4, \u20139;\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne 4, \u20139[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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14. [latex]f\\left(x\\right)=\\frac{3+x}{{x}^{3}-27}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15. [latex]f\\left(x\\right)=\\frac{3x-4}{{x}^{3}-16x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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V.A. at[latex]\\text{ }x=0, 4, -4;\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;[\/latex]Domain is all reals[latex]\\text{ }x\\ne 0,4, \u20134[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]f\\left(x\\right)=\\frac{{x}^{2}-1}{{x}^{3}+9{x}^{2}+14x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17. [latex]f\\left(x\\right)=\\frac{x+5}{{x}^{2}-25}[\/latex]<\/p>\n<\/div>\n

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V.A. at[latex]\\text{ }x=-5;\\text{ }[\/latex]H.A. at[latex]\\text{ }y=0;\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne 5,-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [latex]f\\left(x\\right)=\\frac{x-4}{x-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [latex]f\\left(x\\right)=\\frac{4-2x}{3x-1}[\/latex]<\/p>\n<\/div>\n

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V.A. at[latex]\\text{ }x=\\frac{1}{3};\\text{ }[\/latex]H.A. at[latex]\\text{ }y=-\\frac{2}{3};\\text{ }[\/latex]Domain is all reals[latex]\\text{ }x\\ne \\frac{1}{3}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the x<\/em>– and y<\/em>-intercepts for the functions.<\/p>\n

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20. [latex]f\\left(x\\right)=\\frac{x+5}{{x}^{2}+4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}-x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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none<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22. [latex]f\\left(x\\right)=\\frac{{x}^{2}+8x+7}{{x}^{2}+11x+30}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]f\\left(x\\right)=\\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}[\/latex]<\/p>\n<\/div>\n

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

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25. [latex]f\\left(x\\right)=\\frac{94-2{x}^{2}}{3{x}^{2}-12}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, describe the local and end behavior of the functions.<\/p>\n

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26. [latex]f\\left(x\\right)=\\frac{x}{2x+1}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Local behavior:[latex]\\text{ }x\\to -{\\frac{1}{2}}^{+},f\\left(x\\right)\\to -\\infty ,x\\to -{\\frac{1}{2}}^{-},f\\left(x\\right)\\to \\infty \\text{ }[\/latex]<\/p>\n

End behavior:[latex]\\text{ }x\\to \u00b1\\infty ,f\\left(x\\right)\\to \\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27. [latex]f\\left(x\\right)=\\frac{2x}{x-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28. [latex]f\\left(x\\right)=\\frac{-2x}{x-6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Local behavior:[latex]\\text{ }x\\to {6}^{+},f\\left(x\\right)\\to -\\infty ,x\\to {6}^{-},f\\left(x\\right)\\to \\infty ,\\text{ }[\/latex]End behavior:[latex]\\text{ }x\\to \u00b1\\infty ,f\\left(x\\right)\\to -2[\/latex]<\/p>\n

29. <\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4x+3}{{x}^{2}-4x-5}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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31. [latex]f\\left(x\\right)=\\frac{2{x}^{2}-32}{6{x}^{2}+13x-5}[\/latex]<\/p>\n<\/div>\n

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Local behavior:[latex]\\text{ }x\\to -{\\frac{1}{3}}^{+},f\\left(x\\right)\\to \\infty ,x\\to -{\\frac{1}{3}}^{-},\\text{ }[\/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 \u00b1\\infty ,[\/latex][latex]f\\left(x\\right)\\to \\frac{1}{3}[\/latex]<\/p>\n

<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the slant asymptote of the functions.<\/p>\n

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32. [latex]f\\left(x\\right)=\\frac{24{x}^{2}+6x}{2x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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33. [latex]f\\left(x\\right)=\\frac{4{x}^{2}-10}{2x-4}[\/latex]<\/p>\n<\/div>\n

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[latex]y=2x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34. [latex]f\\left(x\\right)=\\frac{81{x}^{2}-18}{3x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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35. [latex]f\\left(x\\right)=\\frac{6{x}^{3}-5x}{3{x}^{2}+4}[\/latex]<\/p>\n<\/div>\n

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[latex]y=2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36. [latex]f\\left(x\\right)=\\frac{{x}^{2}+5x+4}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.<\/p>\n

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37. The reciprocal function shifted up two units.<\/p>\n<\/div>\n

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

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38. The reciprocal function shifted down one unit and left three units.<\/p>\n<\/div>\n<\/div>\n

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39. The reciprocal squared function shifted to the right 2 units.<\/p>\n<\/div>\n

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

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40. The reciprocal squared function shifted down 2 units and right 1 unit.<\/p>\n<\/div>\n<\/div>\n

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.<\/p>\n

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41. [latex]p\\left(x\\right)=\\frac{2x-3}{x+4}[\/latex]<\/p>\n<\/div>\n

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[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]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42. [latex]q\\left(x\\right)=\\frac{x-5}{3x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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43. [latex]s\\left(x\\right)=\\frac{4}{{\\left(x-2\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n

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

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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44. [latex]r\\left(x\\right)=\\frac{5}{{\\left(x+1\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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45. [latex]f\\left(x\\right)=\\frac{3{x}^{2}-14x-5}{3{x}^{2}+8x-16}[\/latex]<\/p>\n<\/div>\n

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[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

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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46. [latex]g\\left(x\\right)=\\frac{2{x}^{2}+7x-15}{3{x}^{2}-14+15}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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48. [latex]b\\left(x\\right)=\\frac{{x}^{2}-x-6}{{x}^{2}-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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49. [latex]h\\left(x\\right)=\\frac{2{x}^{2}+ x-1}{x-4}[\/latex]<\/p>\n<\/div>\n

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[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]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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50. [latex]k\\left(x\\right)=\\frac{2{x}^{2}-3x-20}{x-5}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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51. [latex]w\\left(x\\right)=\\frac{\\left(x-1\\right)\\left(x+3\\right)\\left(x-5\\right)}{{\\left(x+2\\right)}^{2}\\left(x-4\\right)}[\/latex]<\/p>\n<\/div>\n

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[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]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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52. [latex]z\\left(x\\right)=\\frac{{\\left(x+2\\right)}^{2}\\left(x-5\\right)}{\\left(x-3\\right)\\left(x+1\\right)\\left(x+4\\right)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, write an equation for a rational function with the given characteristics.<\/p>\n

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53. Vertical asymptotes at[latex]\\text{ }x=5\\text{ }[\/latex]and[latex]\\text{ }x=-5,\\text{ }[\/latex]x<\/em>-intercepts at[latex]\\text{ }\\left(2,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(-1,0\\right),\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,4\\right)[\/latex]<\/p>\n<\/div>\n

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

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54. Vertical asymptotes at[latex]\\text{ }x=-4\\text{ }[\/latex]and[latex]\\text{ }x=-1,\\text{ }[\/latex]x-<\/em>intercepts at[latex]\\text{ }\\left(1,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(5,0\\right),\\text{ }[\/latex]y-<\/em>intercept at[latex]\\text{ }\\left(0,7\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55. Vertical asymptotes at[latex]\\text{ }x=-4\\text{ }[\/latex]and[latex]\\text{ }x=-5,\\text{ }[\/latex]x<\/em>-intercepts at[latex]\\text{ }\\left(4,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(-6,0\\right),\\text{ }[\/latex]Horizontal asymptote at[latex]\\text{ }y=7[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]y=7\\frac{{x}^{2}+2x-24}{{x}^{2}+9x+20}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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56. Vertical asymptotes at[latex]\\text{ }x=-3\\text{ }[\/latex]and[latex]\\text{ }x=6,\\text{ }[\/latex]x<\/em>-intercepts at[latex]\\text{ }\\left(-2,0\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(1,0\\right),\\text{ }[\/latex]Horizontal asymptote at[latex]\\text{ }y=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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57. Vertical asymptote at[latex]\\text{ }x=-1,\\text{ }[\/latex]Double zero at[latex]\\text{ }x=2,\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,2\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]y=\\frac{1}{2}\\frac{{x}^{2}-4x+4}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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58. Vertical asymptote at[latex]\\text{ }x=3,\\text{ }[\/latex]Double zero at[latex]\\text{ }x=1,\\text{ }[\/latex]y<\/em>-intercept at[latex]\\text{ }\\left(0,4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use the graphs to write an equation for the function.<\/p>\n

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59.\u00a0\"Graph<\/span><\/div>\n
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Show Answer<\/span><\/p>\n
[latex]y=4\\frac{x-3}{{x}^{2}-x-12}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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60.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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61.\u00a0\"Graph<\/span><\/div>\n
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Show Answer<\/span><\/p>\n
[latex]y=-9\\frac{x-2}{{x}^{2}-9}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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62.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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63.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]y=\\frac{1}{3}\\frac{{x}^{2}+x-6}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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64.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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65.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]y=-6\\frac{{\\left(x-1\\right)}^{2}}{\\left(x+3\\right){\\left(x-2\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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66.\u00a0\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n
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Numeric<\/h4>\n

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote<\/p>\n

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67. [latex]f\\left(x\\right)=\\frac{1}{x-2}[\/latex]<\/p>\n<\/div>\n

Show Answer<\/span><\/p>\n
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\n\n\n\n\n
\u00a0<\/caption>\n
[latex]x[\/latex]<\/td>\n2.01<\/td>\n2.001<\/td>\n2.0001<\/td>\n1.99<\/td>\n1.999<\/td>\n<\/tr>\n
[latex]y[\/latex]<\/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
\u00a0<\/caption>\n
[latex]x[\/latex]<\/th>\n10<\/th>\n100<\/th>\n1,000<\/th>\n10,000<\/th>\n100,000<\/th>\n<\/tr>\n<\/thead>\n
[latex]y[\/latex]<\/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]\\text{ }x=2,\\text{ }[\/latex]Horizontal asymptote[latex]\\text{ }y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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68. [latex]f\\left(x\\right)=\\frac{x}{x-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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69. [latex]f\\left(x\\right)=\\frac{2x}{x+4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n\n\n\n\n
\u00a0<\/caption>\n
[latex]x[\/latex]<\/td>\n\u20134.1<\/td>\n\u20134.01<\/td>\n\u20134.001<\/td>\n\u20133.99<\/td>\n\u20133.999<\/td>\n<\/tr>\n
[latex]y[\/latex]<\/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
\u00a0<\/caption>\n
[latex]x[\/latex]<\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n
[latex]y[\/latex]<\/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]\\text{ }x=-4,\\text{ }[\/latex]Horizontal asymptote[latex]\\text{ }y=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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70. [latex]f\\left(x\\right)=\\frac{2x}{{\\left(x-3\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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71. [latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}+2x+1}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n\n\n\n\n
\u00a0<\/caption>\n
[latex]x[\/latex]<\/td>\n\u2013.9<\/td>\n\u2013.99<\/td>\n\u2013.999<\/td>\n\u20131.1<\/td>\n\u20131.01<\/td>\n<\/tr>\n
[latex]y[\/latex]<\/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
\u00a0<\/caption>\n
[latex]x[\/latex]<\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n<\/tr>\n
[latex]y[\/latex]<\/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]\\text{ }x=-1,\\text{ }[\/latex]Horizontal asymptote[latex]\\text{ }y=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n

Technology<\/h4>\n

For the following exercises, use a calculator to graph[latex]\\text{ }f\\left(x\\right).\\text{ }[\/latex]Use the graph to solve[latex]\\text{ }f\\left(x\\right)>0.[\/latex]<\/p>\n

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72. [latex]f\\left(x\\right)=\\frac{2}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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73. [latex]f\\left(x\\right)=\\frac{4}{2x-3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]<\/p>\n

\"Graph<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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74. [latex]f\\left(x\\right)=\\frac{2}{\\left(x-1\\right)\\left(x+2\\right)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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75. [latex]f\\left(x\\right)=\\frac{x+2}{\\left(x-1\\right)\\left(x-4\\right)}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

\n

Extensions<\/h4>\n

For the following exercises, identify the removable discontinuity.<\/p>\n

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77. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4}{x-2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(2,4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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78. [latex]f\\left(x\\right)=\\frac{{x}^{3}+1}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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79. [latex]f\\left(x\\right)=\\frac{{x}^{2}+x-6}{x-2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(2,5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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80. [latex]f\\left(x\\right)=\\frac{2{x}^{2}+5x-3}{x+3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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81. [latex]f\\left(x\\right)=\\frac{{x}^{3}+{x}^{2}}{x+1}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(\u20131,\\text{1}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Real-World Applications<\/h4>\n

For the following exercises, express a rational function that describes the situation.<\/p>\n

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82. A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after[latex]\\text{ }t\\text{ }[\/latex]minutes.<\/p>\n<\/div>\n<\/div>\n

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83. A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after[latex]\\text{ }t\\text{ }[\/latex]minutes.<\/p>\n<\/div>\n

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

For the following exercises, use the given rational function to answer the question.<\/p>\n

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84. The concentration[latex]\\text{ }C\\text{ }[\/latex]of a drug in a patient\u2019s bloodstream[latex]\\text{ }t\\text{ }[\/latex]hours after injection in given by[latex]\\text{ }C\\left(t\\right)=\\frac{2t}{3+{t}^{2}}.\\text{ }[\/latex]What happens to the concentration of the drug as[latex]\\text{ }t\\text{ }[\/latex]increases?<\/p>\n<\/div>\n<\/div>\n

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85. The concentration[latex]\\text{ }C\\text{ }[\/latex]of a drug in a patient\u2019s bloodstream[latex]\\text{ }t\\text{ }[\/latex]<\/sub>hours after injection is given by[latex]\\text{ }C\\left(t\\right)=\\frac{100t}{2{t}^{2}+75}.\\text{ }[\/latex]Use a calculator to approximate the time when the concentration is highest.<\/p>\n<\/div>\n

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After about 6.12 hours.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.<\/p>\n

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86. An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let[latex]\\text{ }x\\text{ }[\/latex]= length of the side of the base.<\/p>\n<\/div>\n<\/div>\n

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87. A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents\/ square foot. The material for the sides costs 10 cents\/square foot. The material for the top costs 20 cents\/square foot. Determine the dimensions that will yield minimum cost. Let[latex]\\text{ }x\\text{ }[\/latex]= length of the side of the base.<\/p>\n<\/div>\n

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

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88. A right circular cylinder has volume of 100 cubic inches. Find the radius and height that will yield minimum surface area. Let[latex]\\text{ }x\\text{ }[\/latex]= radius.<\/p>\n<\/div>\n<\/div>\n

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89. A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let[latex]\\text{ }x\\text{ }[\/latex]= radius.<\/p>\n<\/div>\n

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

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90. A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents\/square inch to construct the top and bottom and 1 cent\/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let[latex]\\text{ }x\\text{ }[\/latex]= radius.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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