{"id":2138,"date":"2025-08-04T18:42:24","date_gmt":"2025-08-04T18:42:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2138"},"modified":"2026-01-06T16:45:40","modified_gmt":"2026-01-06T16:45:40","slug":"rational-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-get-stronger\/","title":{"raw":"Rational Functions: Get Stronger","rendered":"Rational Functions: Get Stronger"},"content":{"raw":"

Rational Functions<\/h2>\r\n4.\u00a0Can a graph of a rational function have no vertical asymptote? If so, how?\r\n\r\n5. Can a graph of a rational function have no x<\/em>-intercepts? If so, how?\r\n\r\nFor the following exercises, find the domain of the rational functions.\r\n\r\n7. [latex]f\\left(x\\right)=\\frac{x+1}{{x}^{2}-1}[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=\\frac{{x}^{2}+4x - 3}{{x}^{4}-5{x}^{2}+4}[\/latex]\r\n\r\nFor the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.\r\n\r\n13. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}+5x - 36}[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=\\frac{3x - 4}{{x}^{3}-16x}[\/latex]\r\n\r\nFor the following exercises, find the x<\/em>- and y<\/em>-intercepts for the functions.\r\n\r\n21. [latex]f\\left(x\\right)=\\frac{x}{{x}^{2}-x}[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)=\\frac{{x}^{2}+x+6}{{x}^{2}-10x+24}[\/latex]\r\n\r\nFor the following exercises, describe the local and end behavior of the functions.\r\n\r\n25. [latex]f\\left(x\\right)=\\frac{x}{2x+1}[\/latex]\r\n\r\n27. [latex]f\\left(x\\right)=\\frac{-2x}{x - 6}[\/latex]\r\n\r\nFor the following exercises, find the slant asymptote of the functions.\r\n\r\n31. [latex]f\\left(x\\right)=\\frac{4{x}^{2}-10}{2x - 4}[\/latex]\r\n\r\n33. [latex]f\\left(x\\right)=\\frac{6{x}^{3}-5x}{3{x}^{2}+4}[\/latex]\r\n\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\r\n39. [latex]p\\left(x\\right)=\\frac{2x - 3}{x+4}[\/latex]\r\n\r\n41. [latex]s\\left(x\\right)=\\frac{4}{{\\left(x - 2\\right)}^{2}}[\/latex]\r\n\r\n43. [latex]f\\left(x\\right)=\\frac{3{x}^{2}-14x - 5}{3{x}^{2}+8x - 16}[\/latex]\r\n\r\n45. [latex]a\\left(x\\right)=\\frac{{x}^{2}+2x - 3}{{x}^{2}-1}[\/latex]\r\n\r\n47. [latex]h\\left(x\\right)=\\frac{2{x}^{2}+ x - 1}{x - 4}[\/latex]\r\n\r\nFor the following exercises, write an equation for a rational function with the given characteristics.\r\n\r\n51. Vertical asymptotes at x<\/em> = 5 and x\u00a0<\/em>= \u20135, x<\/em>-intercepts at [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex], y<\/em>-intercept at [latex]\\left(0,4\\right)[\/latex]\r\n\r\n53. Vertical asymptotes at [latex]x=-4[\/latex] and [latex]x=-5[\/latex], x<\/em>-intercepts at [latex]\\left(4,0\\right)[\/latex] and [latex]\\left(-6,0\\right)[\/latex], Horizontal asymptote at [latex]y=7[\/latex]\r\n\r\nFor the following exercises, use the graphs to write an equation for the function.\r\n\r\n57.\r\n\"Graph\r\n\r\n63.\r\n\"Graph\r\n\r\nFor the following exercises, use a calculator to graph [latex]f\\left(x\\right)[\/latex]. Use the graph to solve [latex]f\\left(x\\right)>0[\/latex].\r\n\r\n71. [latex]f\\left(x\\right)=\\frac{4}{2x - 3}[\/latex]\r\n\r\n73. [latex]f\\left(x\\right)=\\frac{x+2}{\\left(x - 1\\right)\\left(x - 4\\right)}[\/latex]\r\n\r\nFor the following exercises, identify the removable discontinuity.\r\n\r\n75. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4}{x - 2}[\/latex]\r\n\r\n77. [latex]f\\left(x\\right)=\\frac{{x}^{2}+x - 6}{x - 2}[\/latex]\r\n\r\n79. [latex]f\\left(x\\right)=\\frac{{x}^{3}+{x}^{2}}{x+1}[\/latex]\r\n\r\nFor the following exercises, express a rational function that describes the situation.\r\n\r\n81. 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 t<\/em>\u00a0minutes.\r\n\r\nFor the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.\r\n\r\n85. 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 x<\/em>\u00a0= length of the side of the base.\r\n

Modeling Using Variation<\/h2>\r\nFor the following exercises, write an equation describing the relationship of the given variables.\r\n\r\n5. y<\/em>\u00a0varies directly as the square of x<\/em>\u00a0and when x<\/em> = 4, y<\/em> = 80\r\n\r\n7. y<\/em>\u00a0varies directly as the cube of x<\/em>\u00a0and when x\u00a0<\/em>= 36, y\u00a0<\/em>= 24.\r\n\r\n11. y<\/em>\u00a0varies inversely as the square of x<\/em>\u00a0and when x\u00a0<\/em>= 3, y\u00a0<\/em>= 2.\r\n\r\n15. y<\/em>\u00a0varies inversely as the cube root of x<\/em>\u00a0and when x\u00a0<\/em>= 64, y\u00a0<\/em>= 5.\r\n\r\n17. y<\/em>\u00a0varies jointly as x<\/em>, z<\/em>, and w<\/em>\u00a0and when x\u00a0<\/em>= 1, z\u00a0<\/em>= 2, w\u00a0<\/em>= 5, then y\u00a0<\/em>= 100.\r\n\r\n19. y<\/em>\u00a0varies jointly as x<\/em>\u00a0and the square root of z<\/em>\u00a0and when x\u00a0<\/em>= 2 and z\u00a0<\/em>= 25, then y\u00a0<\/em>= 100.\r\n\r\n21. y<\/em>\u00a0varies jointly as x <\/em>and z<\/em>\u00a0and inversely as w<\/em>. When x\u00a0<\/em>= 3, z\u00a0<\/em>= 5, and w\u00a0<\/em>= 6, then y\u00a0<\/em>= 10.\r\n\r\nFor the following exercises, use the given information to find the unknown value.\r\n\r\n25. y<\/em>\u00a0varies directly as the square of x<\/em>. When x\u00a0<\/em>= 2, then y\u00a0<\/em>= 16. Find y<\/em>\u00a0when x\u00a0<\/em>= 8.\r\n\r\n29. y<\/i>\u00a0varies inversely with x<\/em>. When x\u00a0<\/em>= 3, then y\u00a0<\/em>= 2. Find y<\/em>\u00a0when x\u00a0<\/em>= 1.\r\n\r\n35. y<\/em>\u00a0varies jointly as x<\/i>, z<\/em>, and w<\/em>. When x\u00a0<\/em>= 2, z\u00a0<\/em>= 1, and w\u00a0<\/em>= 12, then y\u00a0<\/em>= 72. Find y<\/em>\u00a0when x\u00a0<\/em>= 1, z\u00a0<\/em>= 2,\u00a0and w\u00a0<\/em>= 3.\r\n\r\n39. y<\/em>\u00a0varies jointly as the square of x<\/em>\u00a0and the cube of z<\/em>\u00a0and inversely as the square root of w<\/em>.\u00a0When x\u00a0<\/em>= 2, z\u00a0<\/em>= 2, and w\u00a0<\/em>= 64, then y\u00a0<\/em>= 12. Find y<\/em>\u00a0when x\u00a0<\/em>= 1, z\u00a0<\/em>= 3, and w\u00a0<\/em>= 4.\r\n\r\nFor the following exercises, use the given information to answer the questions.\r\n\r\n51. The distance s<\/em>\u00a0that an object falls varies directly with the square of the time, t<\/em>, of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?\r\n\r\n53. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?","rendered":"

Rational Functions<\/h2>\n

4.\u00a0Can a graph of a rational function have no vertical asymptote? If so, how?<\/p>\n

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

51. Vertical asymptotes at x<\/em> = 5 and x\u00a0<\/em>= \u20135, x<\/em>-intercepts at [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex], y<\/em>-intercept at [latex]\\left(0,4\\right)[\/latex]<\/p>\n

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

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

57.
\n\"Graph<\/p>\n

63.
\n\"Graph<\/p>\n

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

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

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

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

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

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

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

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

81. 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 t<\/em>\u00a0minutes.<\/p>\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

85. 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 x<\/em>\u00a0= length of the side of the base.<\/p>\n

Modeling Using Variation<\/h2>\n

For the following exercises, write an equation describing the relationship of the given variables.<\/p>\n

5. y<\/em>\u00a0varies directly as the square of x<\/em>\u00a0and when x<\/em> = 4, y<\/em> = 80<\/p>\n

7. y<\/em>\u00a0varies directly as the cube of x<\/em>\u00a0and when x\u00a0<\/em>= 36, y\u00a0<\/em>= 24.<\/p>\n

11. y<\/em>\u00a0varies inversely as the square of x<\/em>\u00a0and when x\u00a0<\/em>= 3, y\u00a0<\/em>= 2.<\/p>\n

15. y<\/em>\u00a0varies inversely as the cube root of x<\/em>\u00a0and when x\u00a0<\/em>= 64, y\u00a0<\/em>= 5.<\/p>\n

17. y<\/em>\u00a0varies jointly as x<\/em>, z<\/em>, and w<\/em>\u00a0and when x\u00a0<\/em>= 1, z\u00a0<\/em>= 2, w\u00a0<\/em>= 5, then y\u00a0<\/em>= 100.<\/p>\n

19. y<\/em>\u00a0varies jointly as x<\/em>\u00a0and the square root of z<\/em>\u00a0and when x\u00a0<\/em>= 2 and z\u00a0<\/em>= 25, then y\u00a0<\/em>= 100.<\/p>\n

21. y<\/em>\u00a0varies jointly as x <\/em>and z<\/em>\u00a0and inversely as w<\/em>. When x\u00a0<\/em>= 3, z\u00a0<\/em>= 5, and w\u00a0<\/em>= 6, then y\u00a0<\/em>= 10.<\/p>\n

For the following exercises, use the given information to find the unknown value.<\/p>\n

25. y<\/em>\u00a0varies directly as the square of x<\/em>. When x\u00a0<\/em>= 2, then y\u00a0<\/em>= 16. Find y<\/em>\u00a0when x\u00a0<\/em>= 8.<\/p>\n

29. y<\/i>\u00a0varies inversely with x<\/em>. When x\u00a0<\/em>= 3, then y\u00a0<\/em>= 2. Find y<\/em>\u00a0when x\u00a0<\/em>= 1.<\/p>\n

35. y<\/em>\u00a0varies jointly as x<\/i>, z<\/em>, and w<\/em>. When x\u00a0<\/em>= 2, z\u00a0<\/em>= 1, and w\u00a0<\/em>= 12, then y\u00a0<\/em>= 72. Find y<\/em>\u00a0when x\u00a0<\/em>= 1, z\u00a0<\/em>= 2,\u00a0and w\u00a0<\/em>= 3.<\/p>\n

39. y<\/em>\u00a0varies jointly as the square of x<\/em>\u00a0and the cube of z<\/em>\u00a0and inversely as the square root of w<\/em>.\u00a0When x\u00a0<\/em>= 2, z\u00a0<\/em>= 2, and w\u00a0<\/em>= 64, then y\u00a0<\/em>= 12. Find y<\/em>\u00a0when x\u00a0<\/em>= 1, z\u00a0<\/em>= 3, and w\u00a0<\/em>= 4.<\/p>\n

For the following exercises, use the given information to answer the questions.<\/p>\n

51. The distance s<\/em>\u00a0that an object falls varies directly with the square of the time, t<\/em>, of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?<\/p>\n

53. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?<\/p>\n","protected":false},"author":67,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2138"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2138\/revisions"}],"predecessor-version":[{"id":5206,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2138\/revisions\/5206"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}