{"id":131,"date":"2026-01-12T15:49:55","date_gmt":"2026-01-12T15:49:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=131"},"modified":"2026-01-12T15:49:55","modified_gmt":"2026-01-12T15:49:55","slug":"rational-and-radical-functions-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/rational-and-radical-functions-get-stronger-answer-key\/","title":{"raw":"Rational and Radical Functions: Get Stronger Answer Key","rendered":"Rational and Radical Functions: Get Stronger Answer Key"},"content":{"raw":"

Rational Functions<\/span><\/h2>\r\n
    \r\n \t
  1. All reals [latex]x \\neq 1[\/latex], [latex]1[\/latex]<\/li>\r\n \t
  2. All reals [latex]x \\neq -1, -2, 1, 2[\/latex]<\/li>\r\n \t
  3. V.A. at [latex]x=-\\dfrac{2}{5}[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq -\\dfrac{2}{5}[\/latex]<\/li>\r\n \t
  4. V.A. at [latex]x=4, -9[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq 4, -9[\/latex]<\/li>\r\n \t
  5. V.A. at [latex]x=0, 4, -4[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq 0, 4, -4[\/latex]<\/li>\r\n \t
  6. V.A. at [latex]x=5[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq 5, -5[\/latex]<\/li>\r\n \t
  7. V.A. at [latex]x=\\dfrac{1}{3}[\/latex]; H.A. at [latex]y=-\\dfrac{2}{3}[\/latex]; Domain is all reals [latex]x \\neq \\dfrac{1}{3}[\/latex]<\/li>\r\n \t
  8. none<\/li>\r\n \t
  9. [latex]x[\/latex]-intercepts none, [latex]y[\/latex]-intercept [latex](0, \\dfrac{1}{4})[\/latex]<\/li>\r\n \t
  10. Local behavior: [latex]x \\to -\\dfrac{1}{2}^+[\/latex], [latex]f(x) \\to -\\infty[\/latex], [latex]x \\to -\\dfrac{1}{2}^-[\/latex], [latex]f(x) \\to \\infty[\/latex] End behavior: [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to \\dfrac{1}{2}[\/latex]<\/li>\r\n \t
  11. Local behavior: [latex]x \\to 6^+[\/latex], [latex]f(x) \\to -\\infty[\/latex], [latex]x \\to 6^-[\/latex], [latex]f(x) \\to \\infty[\/latex], End behavior: [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to -2[\/latex]<\/li>\r\n \t
  12. Local behavior: [latex]x \\to \\dfrac{1}{3}^+[\/latex], [latex]f(x) \\to -\\infty[\/latex], [latex]x \\to \\dfrac{1}{3}^-[\/latex], [latex]f(x) \\to \\infty[\/latex], [latex]x \\to \\dfrac{5}{2}^+[\/latex], [latex]f(x) \\to \\infty[\/latex], [latex]x \\to \\dfrac{5}{2}^-[\/latex], [latex]f(x) \\to -\\infty[\/latex] End behavior: [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to \\dfrac{1}{3}[\/latex]<\/li>\r\n \t
  13. [latex]y = 2x + 4[\/latex]<\/li>\r\n \t
  14. [latex]y = -2x[\/latex]<\/li>\r\n \t
  15. V.A. [latex]x=-4[\/latex], H.A. [latex]y=2[\/latex]; [latex](\\dfrac{3}{2},0)[\/latex]; [latex](0,-\\dfrac{3}{4})[\/latex]\r\n\"Graph<\/li>\r\n \t
  16. V.A. [latex]x=2[\/latex], H.A. [latex]y=0[\/latex], [latex](0,1)[\/latex]\r\n\"Graph<\/li>\r\n \t
  17. V.A. [latex]x=-4[\/latex], [latex]x=\\dfrac{4}{3}[\/latex], H.A. [latex]y=1[\/latex]; [latex](5,0)[\/latex]; [latex](-\\dfrac{1}{3},0)[\/latex]; [latex](0,\\dfrac{5}{16})[\/latex]\r\n\"Graph<\/li>\r\n \t
  18. V.A. [latex]x=-1[\/latex], H.A. [latex]y=1[\/latex]; [latex](-3,0)[\/latex]; [latex](0,3)[\/latex]\r\n\"Graph<\/li>\r\n \t
  19. [latex]y = 50\\dfrac{x^2-x-2}{x^2-25}[\/latex]<\/li>\r\n \t
  20. [latex]y = 7\\dfrac{x^2+2x-24}{x^2+9x+20}[\/latex]<\/li>\r\n \t
  21. [latex]y = \\dfrac{1}{2}\\dfrac{x^2-4x+4}{x+1}[\/latex]<\/li>\r\n \t
  22. [latex]y = 4\\dfrac{x-3}{x^2-x-12}[\/latex]<\/li>\r\n \t
  23. [latex]y = \\dfrac{27(x-2)}{(x+3)(x-3)^2}[\/latex]<\/li>\r\n \t
  24. [latex]C(t) = \\dfrac{8+2t}{300+20t}[\/latex]<\/li>\r\n \t
  25. After about [latex]6.12[\/latex] hours.<\/li>\r\n \t
  26. [latex]A(x) = 50x^2 + \\dfrac{800}{x}[\/latex]. [latex]2[\/latex] by [latex]2[\/latex] by [latex]5[\/latex] feet.<\/li>\r\n \t
  27. [latex]A(x) = \\pi x^2 + \\dfrac{100}{x}[\/latex]. Radius = [latex]2.52[\/latex] meters.<\/li>\r\n<\/ol>\r\n

    Inverses and Radical Functions<\/span><\/h2>\r\n
      \r\n \t
    1. [latex]f^{-1}(x) = \\sqrt{x + 4}[\/latex]<\/li>\r\n \t
    2. [latex]f^{-1}(x) = \\sqrt{x + 3} - 1[\/latex]<\/li>\r\n \t
    3. [latex]f^{-1}(x) = \\sqrt{12 - x}[\/latex]<\/li>\r\n \t
    4. [latex]f^{-1}(x) = \\pm\\sqrt{\\dfrac{x-4}{2}}[\/latex]<\/li>\r\n \t
    5. [latex]f^{-1}(x) = \\sqrt{\\dfrac{x-1}{3}-3}[\/latex]<\/li>\r\n \t
    6. [latex]f^{-1}(x) = \\sqrt{\\dfrac{4-x}{2}-3}[\/latex]<\/li>\r\n \t
    7. [latex]f^{-1}(x) = \\dfrac{3-x^2}{4}[\/latex], [latex][0,\\infty)[\/latex]<\/li>\r\n \t
    8. [latex]f^{-1}(x) = \\dfrac{(x-5)^2+8}{6}[\/latex]<\/li>\r\n \t
    9. [latex]f^{-1}(x) = (3 - x)^2[\/latex]<\/li>\r\n \t
    10. [latex]f^{-1}(x) = \\dfrac{4x+3}{7}[\/latex]<\/li>\r\n \t
    11. [latex]f^{-1}(x) = \\dfrac{7x-3}{1-x}[\/latex]<\/li>\r\n \t
    12. [latex]f^{-1}(x) = \\dfrac{2x-1}{5x+5}[\/latex]<\/li>\r\n \t
    13. [latex]f^{-1}(x) = \\sqrt{x + 3} - 2[\/latex]<\/li>\r\n \t
    14. [latex]f^{-1}(x) = \\sqrt{x - 2}[\/latex]\r\n\"Graph<\/li>\r\n \t
    15. [latex]f^{-1}(x) = \\sqrt{x - 3}[\/latex]\r\n\"Graph<\/li>\r\n \t
    16. [latex]f^{-1}(x) = \\sqrt[3]{x - 3}[\/latex]\r\n\"<\/li>\r\n \t
    17. [latex][-1,0] \\cup [1,\\infty)[\/latex]\r\n\"Graph<\/li>\r\n \t
    18. [latex][-3,0] \\cup (4,\\infty)[\/latex]\r\n\"Graph<\/li>\r\n \t
    19. [latex][-\\infty,-4] \\cdot [-3,3][\/latex]\r\n\"Graph<\/li>\r\n \t
    20. [latex]t(h) = \\sqrt{\\dfrac{600-h}{16}}[\/latex], [latex]3.54[\/latex] seconds<\/li>\r\n \t
    21. [latex]r(A) = \\sqrt{\\dfrac{A}{4\\pi}}[\/latex], [latex]\\approx 8.92[\/latex] in.<\/li>\r\n \t
    22. [latex]l(T) = 32.2\\left(\\dfrac{T}{2\\pi}\\right)[\/latex], [latex]\\approx 3.26 [\/latex]ft<\/li>\r\n \t
    23. [latex]r(A) = \\sqrt{\\dfrac{A+8\\pi}{2\\pi}}-2[\/latex], [latex]3.99[\/latex] ft<\/li>\r\n \t
    24. [latex]r(V) = \\sqrt{\\dfrac{V}{10\\pi}}[\/latex], [latex]\\approx 5.64[\/latex] ft<\/li>\r\n<\/ol>\r\n

      Variations<\/span><\/h2>\r\n
        \r\n \t
      1. [latex]y = 5x^2[\/latex]<\/li>\r\n \t
      2. [latex]y = \\dfrac{1}{1944}x^3[\/latex]<\/li>\r\n \t
      3. [latex]y = 6x^4[\/latex]<\/li>\r\n \t
      4. [latex]y = \\dfrac{18}{x^2}[\/latex]<\/li>\r\n \t
      5. [latex]y = \\dfrac{81}{x^4}[\/latex]<\/li>\r\n \t
      6. [latex]y = \\dfrac{20}{\\sqrt[3]{x}}[\/latex]<\/li>\r\n \t
      7. [latex]y = 10xzw[\/latex]<\/li>\r\n \t
      8. [latex]y = 10x\\sqrt{z}[\/latex]<\/li>\r\n \t
      9. [latex]y = 4\\dfrac{xz}{w}[\/latex]<\/li>\r\n \t
      10. [latex]y = 40\\dfrac{xz}{\\sqrt{wt^2}}[\/latex]<\/li>\r\n \t
      11. [latex]y = 256[\/latex]<\/li>\r\n \t
      12. [latex]y = 6[\/latex]<\/li>\r\n \t
      13. [latex]y = 6[\/latex]<\/li>\r\n \t
      14. [latex]y = 27[\/latex]<\/li>\r\n \t
      15. [latex]y = 3[\/latex]<\/li>\r\n \t
      16. [latex]y = 18[\/latex]<\/li>\r\n \t
      17. [latex]y = 90[\/latex]<\/li>\r\n \t
      18. [latex]y = \\dfrac{81}{2}[\/latex]<\/li>\r\n \t
      19. [latex]3[\/latex] seconds<\/li>\r\n \t
      20. [latex]48[\/latex] inches<\/li>\r\n \t
      21. [latex]49.75[\/latex] pounds<\/li>\r\n \t
      22. [latex]33.33[\/latex] amperes<\/li>\r\n \t
      23. [latex]2.88[\/latex] inches<\/li>\r\n<\/ol>","rendered":"

        Rational Functions<\/span><\/h2>\n
          \n
        1. All reals [latex]x \\neq 1[\/latex], [latex]1[\/latex]<\/li>\n
        2. All reals [latex]x \\neq -1, -2, 1, 2[\/latex]<\/li>\n
        3. V.A. at [latex]x=-\\dfrac{2}{5}[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq -\\dfrac{2}{5}[\/latex]<\/li>\n
        4. V.A. at [latex]x=4, -9[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq 4, -9[\/latex]<\/li>\n
        5. V.A. at [latex]x=0, 4, -4[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq 0, 4, -4[\/latex]<\/li>\n
        6. V.A. at [latex]x=5[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x \\neq 5, -5[\/latex]<\/li>\n
        7. V.A. at [latex]x=\\dfrac{1}{3}[\/latex]; H.A. at [latex]y=-\\dfrac{2}{3}[\/latex]; Domain is all reals [latex]x \\neq \\dfrac{1}{3}[\/latex]<\/li>\n
        8. none<\/li>\n
        9. [latex]x[\/latex]-intercepts none, [latex]y[\/latex]-intercept [latex](0, \\dfrac{1}{4})[\/latex]<\/li>\n
        10. Local behavior: [latex]x \\to -\\dfrac{1}{2}^+[\/latex], [latex]f(x) \\to -\\infty[\/latex], [latex]x \\to -\\dfrac{1}{2}^-[\/latex], [latex]f(x) \\to \\infty[\/latex] End behavior: [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to \\dfrac{1}{2}[\/latex]<\/li>\n
        11. Local behavior: [latex]x \\to 6^+[\/latex], [latex]f(x) \\to -\\infty[\/latex], [latex]x \\to 6^-[\/latex], [latex]f(x) \\to \\infty[\/latex], End behavior: [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to -2[\/latex]<\/li>\n
        12. Local behavior: [latex]x \\to \\dfrac{1}{3}^+[\/latex], [latex]f(x) \\to -\\infty[\/latex], [latex]x \\to \\dfrac{1}{3}^-[\/latex], [latex]f(x) \\to \\infty[\/latex], [latex]x \\to \\dfrac{5}{2}^+[\/latex], [latex]f(x) \\to \\infty[\/latex], [latex]x \\to \\dfrac{5}{2}^-[\/latex], [latex]f(x) \\to -\\infty[\/latex] End behavior: [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to \\dfrac{1}{3}[\/latex]<\/li>\n
        13. [latex]y = 2x + 4[\/latex]<\/li>\n
        14. [latex]y = -2x[\/latex]<\/li>\n
        15. V.A. [latex]x=-4[\/latex], H.A. [latex]y=2[\/latex]; [latex](\\dfrac{3}{2},0)[\/latex]; [latex](0,-\\dfrac{3}{4})[\/latex]
          \n\"Graph<\/li>\n
        16. V.A. [latex]x=2[\/latex], H.A. [latex]y=0[\/latex], [latex](0,1)[\/latex]
          \n\"Graph<\/li>\n
        17. V.A. [latex]x=-4[\/latex], [latex]x=\\dfrac{4}{3}[\/latex], H.A. [latex]y=1[\/latex]; [latex](5,0)[\/latex]; [latex](-\\dfrac{1}{3},0)[\/latex]; [latex](0,\\dfrac{5}{16})[\/latex]
          \n\"Graph<\/li>\n
        18. V.A. [latex]x=-1[\/latex], H.A. [latex]y=1[\/latex]; [latex](-3,0)[\/latex]; [latex](0,3)[\/latex]
          \n\"Graph<\/li>\n
        19. [latex]y = 50\\dfrac{x^2-x-2}{x^2-25}[\/latex]<\/li>\n
        20. [latex]y = 7\\dfrac{x^2+2x-24}{x^2+9x+20}[\/latex]<\/li>\n
        21. [latex]y = \\dfrac{1}{2}\\dfrac{x^2-4x+4}{x+1}[\/latex]<\/li>\n
        22. [latex]y = 4\\dfrac{x-3}{x^2-x-12}[\/latex]<\/li>\n
        23. [latex]y = \\dfrac{27(x-2)}{(x+3)(x-3)^2}[\/latex]<\/li>\n
        24. [latex]C(t) = \\dfrac{8+2t}{300+20t}[\/latex]<\/li>\n
        25. After about [latex]6.12[\/latex] hours.<\/li>\n
        26. [latex]A(x) = 50x^2 + \\dfrac{800}{x}[\/latex]. [latex]2[\/latex] by [latex]2[\/latex] by [latex]5[\/latex] feet.<\/li>\n
        27. [latex]A(x) = \\pi x^2 + \\dfrac{100}{x}[\/latex]. Radius = [latex]2.52[\/latex] meters.<\/li>\n<\/ol>\n

          Inverses and Radical Functions<\/span><\/h2>\n
            \n
          1. [latex]f^{-1}(x) = \\sqrt{x + 4}[\/latex]<\/li>\n
          2. [latex]f^{-1}(x) = \\sqrt{x + 3} - 1[\/latex]<\/li>\n
          3. [latex]f^{-1}(x) = \\sqrt{12 - x}[\/latex]<\/li>\n
          4. [latex]f^{-1}(x) = \\pm\\sqrt{\\dfrac{x-4}{2}}[\/latex]<\/li>\n
          5. [latex]f^{-1}(x) = \\sqrt{\\dfrac{x-1}{3}-3}[\/latex]<\/li>\n
          6. [latex]f^{-1}(x) = \\sqrt{\\dfrac{4-x}{2}-3}[\/latex]<\/li>\n
          7. [latex]f^{-1}(x) = \\dfrac{3-x^2}{4}[\/latex], [latex][0,\\infty)[\/latex]<\/li>\n
          8. [latex]f^{-1}(x) = \\dfrac{(x-5)^2+8}{6}[\/latex]<\/li>\n
          9. [latex]f^{-1}(x) = (3 - x)^2[\/latex]<\/li>\n
          10. [latex]f^{-1}(x) = \\dfrac{4x+3}{7}[\/latex]<\/li>\n
          11. [latex]f^{-1}(x) = \\dfrac{7x-3}{1-x}[\/latex]<\/li>\n
          12. [latex]f^{-1}(x) = \\dfrac{2x-1}{5x+5}[\/latex]<\/li>\n
          13. [latex]f^{-1}(x) = \\sqrt{x + 3} - 2[\/latex]<\/li>\n
          14. [latex]f^{-1}(x) = \\sqrt{x - 2}[\/latex]
            \n\"Graph<\/li>\n
          15. [latex]f^{-1}(x) = \\sqrt{x - 3}[\/latex]
            \n\"Graph<\/li>\n
          16. [latex]f^{-1}(x) = \\sqrt[3]{x - 3}[\/latex]
            \n\"Graph<\/li>\n
          17. [latex][-1,0] \\cup [1,\\infty)[\/latex]
            \n\"Graph<\/li>\n
          18. [latex][-3,0] \\cup (4,\\infty)[\/latex]
            \n\"Graph<\/li>\n
          19. [latex][-\\infty,-4] \\cdot [-3,3][\/latex]
            \n\"Graph<\/li>\n
          20. [latex]t(h) = \\sqrt{\\dfrac{600-h}{16}}[\/latex], [latex]3.54[\/latex] seconds<\/li>\n
          21. [latex]r(A) = \\sqrt{\\dfrac{A}{4\\pi}}[\/latex], [latex]\\approx 8.92[\/latex] in.<\/li>\n
          22. [latex]l(T) = 32.2\\left(\\dfrac{T}{2\\pi}\\right)[\/latex], [latex]\\approx 3.26[\/latex]ft<\/li>\n
          23. [latex]r(A) = \\sqrt{\\dfrac{A+8\\pi}{2\\pi}}-2[\/latex], [latex]3.99[\/latex] ft<\/li>\n
          24. [latex]r(V) = \\sqrt{\\dfrac{V}{10\\pi}}[\/latex], [latex]\\approx 5.64[\/latex] ft<\/li>\n<\/ol>\n

            Variations<\/span><\/h2>\n
              \n
            1. [latex]y = 5x^2[\/latex]<\/li>\n
            2. [latex]y = \\dfrac{1}{1944}x^3[\/latex]<\/li>\n
            3. [latex]y = 6x^4[\/latex]<\/li>\n
            4. [latex]y = \\dfrac{18}{x^2}[\/latex]<\/li>\n
            5. [latex]y = \\dfrac{81}{x^4}[\/latex]<\/li>\n
            6. [latex]y = \\dfrac{20}{\\sqrt[3]{x}}[\/latex]<\/li>\n
            7. [latex]y = 10xzw[\/latex]<\/li>\n
            8. [latex]y = 10x\\sqrt{z}[\/latex]<\/li>\n
            9. [latex]y = 4\\dfrac{xz}{w}[\/latex]<\/li>\n
            10. [latex]y = 40\\dfrac{xz}{\\sqrt{wt^2}}[\/latex]<\/li>\n
            11. [latex]y = 256[\/latex]<\/li>\n
            12. [latex]y = 6[\/latex]<\/li>\n
            13. [latex]y = 6[\/latex]<\/li>\n
            14. [latex]y = 27[\/latex]<\/li>\n
            15. [latex]y = 3[\/latex]<\/li>\n
            16. [latex]y = 18[\/latex]<\/li>\n
            17. [latex]y = 90[\/latex]<\/li>\n
            18. [latex]y = \\dfrac{81}{2}[\/latex]<\/li>\n
            19. [latex]3[\/latex] seconds<\/li>\n
            20. [latex]48[\/latex] inches<\/li>\n
            21. [latex]49.75[\/latex] pounds<\/li>\n
            22. [latex]33.33[\/latex] amperes<\/li>\n
            23. [latex]2.88[\/latex] inches<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":106,"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\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/131"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/131\/revisions"}],"predecessor-version":[{"id":139,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/131\/revisions\/139"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/106"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/131\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=131"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=131"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=131"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}