{"id":292,"date":"2026-01-30T23:00:20","date_gmt":"2026-01-30T23:00:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/rational-functions-get-stronger-key-precalculus-practice-page-answer-keys\/"},"modified":"2026-08-07T16:52:10","modified_gmt":"2026-08-07T16:52:10","slug":"rational-functions-get-stronger-key-precalculus-practice-page-answer-keys","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/rational-functions-get-stronger-key-precalculus-practice-page-answer-keys\/","title":{"raw":"Rational Functions: Get Stronger Key -- Precalculus Practice Page Answer Keys","rendered":"Rational Functions: Get Stronger Key &#8212; Precalculus Practice Page Answer Keys"},"content":{"raw":"<h2>Rational Functions Solutions<\/h2>\r\n2.\u00a0Yes. The numerator of the formula of the functions would have only complex roots and\/or factors common to both the numerator and denominator.\r\n\r\n3.\u00a0[latex]\\text{All reals }x\\ne -1, 1[\/latex]\r\n\r\n4.\u00a0[latex]\\text{All reals }x\\ne -1, -2, 1, 2[\/latex]\r\n\r\n5.\u00a0V.A. at [latex]x=4, -9[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 4, -9[\/latex]\r\n\r\n6.\u00a0V.A. at [latex]x=0, 4, -4[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 0,4, -4[\/latex]\r\n\r\n7.\u00a0none\r\n\r\n8.\u00a0[latex]x\\text{-intercepts none, }y\\text{-intercept }\\left(0,\\frac{1}{4}\\right)[\/latex]\r\n\r\n9.\u00a0Local behavior: [latex]x\\to -{\\frac{1}{2}}^{+},f\\left(x\\right)\\to -\\infty ,x\\to -{\\frac{1}{2}}^{-},f\\left(x\\right)\\to \\infty[\/latex]\r\n\r\nEnd behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to \\frac{1}{2}[\/latex]\r\n\r\n10.\u00a0Local behavior: [latex]x\\to {6}^{+},f\\left(x\\right)\\to -\\infty ,x\\to {6}^{-},f\\left(x\\right)\\to \\infty[\/latex], End behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to -2[\/latex]\r\n\r\n11.\u00a0[latex]y=2x+4[\/latex]\r\n\r\n12.\u00a0[latex]y=2x[\/latex]\r\n\r\n13.\u00a0[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]\r\n<img src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230016\/CNX_Precalc_Figure_03_07_205.jpg\" alt=\"Graph of p(x)=(2x-3)\/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.\" \/>\r\n\r\n14.\u00a0[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0,\\text{ }\\left(0,1\\right)[\/latex]\r\n<img src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230017\/CNX_Precalc_Figure_03_07_207.jpg\" alt=\"Graph of s(x)=4\/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.\" \/>\r\n\r\n15.\u00a0[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]\r\n\r\n16.\u00a0[latex]V.A.\\text{ }x=-1,\\text{ }H.A.\\text{ }y=1;\\left(-3,0\\right);\\left(0,3\\right)[\/latex]\r\n<img src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230017\/CNX_Precalc_Figure_03_07_209.jpg\" alt=\"Graph of f(x)=(3x^2-14x-5)\/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4\/3 and horizontal asymptote at y=1.\" \/>\r\n\r\n17.\u00a0[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]\r\n<img src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230018\/CNX_Precalc_Figure_03_07_213.jpg\" alt=\"Graph of h(x)=(2x^2+x-1)\/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.\" \/>\r\n\r\n18.\u00a0[latex]y=50\\frac{{x}^{2}-x - 2}{{x}^{2}-25}[\/latex]\r\n\r\n19.\u00a0[latex]y=7\\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20}[\/latex]\r\n\r\n20.\u00a0[latex]y=4\\frac{x - 3}{{x}^{2}-x - 12}[\/latex]\r\n\r\n21.\u00a0[latex]y=\\frac{1}{3}\\frac{{x}^{2}+x - 6}{x - 1}[\/latex]\r\n\r\n22.\u00a0[latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]\r\n<img src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230019\/CNX_Precalc_Figure_03_07_226.jpg\" alt=\"Graph of f(x)=4\/(2x-3).\" \/>\r\n\r\n23.\u00a0[latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]\r\n<img src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230019\/CNX_Precalc_Figure_03_07_228.jpg\" alt=\"Graph of f(x)=(x+2)\/(x-1)(x-4).\" \/>\r\n\r\n24.\u00a0[latex]\\left(2,4\\right)[\/latex]\r\n\r\n25.\u00a0[latex]\\left(2,5\\right)[\/latex]\r\n\r\n26.\u00a0[latex]\\left(-1,\\text{1}\\right)[\/latex]\r\n\r\n27.\u00a0[latex]C\\left(t\\right)=\\frac{8+2t}{300+20t}[\/latex]\r\n\r\n28.\u00a0[latex]A\\left(x\\right)=50{x}^{2}+\\frac{800}{x}[\/latex]. 2 by 2 by 5 feet.\r\n<h2>Modeling Using Variation Solutions<\/h2>\r\n1.\u00a0[latex]y=5{x}^{2}[\/latex]\r\n\r\n2.\u00a0[latex]y=10{x}^{3}[\/latex]\r\n\r\n3.\u00a0[latex]y=\\frac{18}{{x}^{2}}[\/latex]\r\n\r\n4.\u00a0[latex]y=\\frac{20}{\\sqrt[3]{x}}[\/latex]\r\n\r\n5.\u00a0[latex]y=10xzw[\/latex]\r\n\r\n6.\u00a0[latex]y=10x\\sqrt{z}[\/latex]\r\n\r\n7.\u00a0[latex]y=4\\frac{xz}{w}[\/latex]\r\n\r\n8.\u00a0[latex]y=256[\/latex]\r\n\r\n9.\u00a0[latex]y=6[\/latex]\r\n\r\n10.\u00a0[latex]y=18[\/latex]\r\n\r\n11.\u00a0[latex]y=\\frac{81}{2}[\/latex]\r\n\r\n12.\u00a03 seconds\r\n\r\n13.\u00a048 inches\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<h2>Rational Functions Solutions<\/h2>\n<p>2.\u00a0Yes. 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<p>3.\u00a0[latex]\\text{All reals }x\\ne -1, 1[\/latex]<\/p>\n<p>4.\u00a0[latex]\\text{All reals }x\\ne -1, -2, 1, 2[\/latex]<\/p>\n<p>5.\u00a0V.A. at [latex]x=4, -9[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 4, -9[\/latex]<\/p>\n<p>6.\u00a0V.A. at [latex]x=0, 4, -4[\/latex]; H.A. at [latex]y=0[\/latex]; Domain is all reals [latex]x\\ne 0,4, -4[\/latex]<\/p>\n<p>7.\u00a0none<\/p>\n<p>8.\u00a0[latex]x\\text{-intercepts none, }y\\text{-intercept }\\left(0,\\frac{1}{4}\\right)[\/latex]<\/p>\n<p>9.\u00a0Local behavior: [latex]x\\to -{\\frac{1}{2}}^{+},f\\left(x\\right)\\to -\\infty ,x\\to -{\\frac{1}{2}}^{-},f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p>End behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to \\frac{1}{2}[\/latex]<\/p>\n<p>10.\u00a0Local behavior: [latex]x\\to {6}^{+},f\\left(x\\right)\\to -\\infty ,x\\to {6}^{-},f\\left(x\\right)\\to \\infty[\/latex], End behavior: [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to -2[\/latex]<\/p>\n<p>11.\u00a0[latex]y=2x+4[\/latex]<\/p>\n<p>12.\u00a0[latex]y=2x[\/latex]<\/p>\n<p>13.\u00a0[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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230016\/CNX_Precalc_Figure_03_07_205.jpg\" alt=\"Graph of p(x)=(2x-3)\/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.\" \/><\/p>\n<p>14.\u00a0[latex]V.A.\\text{ }x=2,\\text{ }H.A.\\text{ }y=0,\\text{ }\\left(0,1\\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230017\/CNX_Precalc_Figure_03_07_207.jpg\" alt=\"Graph of s(x)=4\/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.\" \/><\/p>\n<p>15.\u00a0[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<p>16.\u00a0[latex]V.A.\\text{ }x=-1,\\text{ }H.A.\\text{ }y=1;\\left(-3,0\\right);\\left(0,3\\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230017\/CNX_Precalc_Figure_03_07_209.jpg\" alt=\"Graph of f(x)=(3x^2-14x-5)\/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4\/3 and horizontal asymptote at y=1.\" \/><\/p>\n<p>17.\u00a0[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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230018\/CNX_Precalc_Figure_03_07_213.jpg\" alt=\"Graph of h(x)=(2x^2+x-1)\/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.\" \/><\/p>\n<p>18.\u00a0[latex]y=50\\frac{{x}^{2}-x - 2}{{x}^{2}-25}[\/latex]<\/p>\n<p>19.\u00a0[latex]y=7\\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20}[\/latex]<\/p>\n<p>20.\u00a0[latex]y=4\\frac{x - 3}{{x}^{2}-x - 12}[\/latex]<\/p>\n<p>21.\u00a0[latex]y=\\frac{1}{3}\\frac{{x}^{2}+x - 6}{x - 1}[\/latex]<\/p>\n<p>22.\u00a0[latex]\\left(\\frac{3}{2},\\infty \\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230019\/CNX_Precalc_Figure_03_07_226.jpg\" alt=\"Graph of f(x)=4\/(2x-3).\" \/><\/p>\n<p>23.\u00a0[latex]\\left(-2,1\\right)\\cup \\left(4,\\infty \\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/60\/2026\/01\/30230019\/CNX_Precalc_Figure_03_07_228.jpg\" alt=\"Graph of f(x)=(x+2)\/(x-1)(x-4).\" \/><\/p>\n<p>24.\u00a0[latex]\\left(2,4\\right)[\/latex]<\/p>\n<p>25.\u00a0[latex]\\left(2,5\\right)[\/latex]<\/p>\n<p>26.\u00a0[latex]\\left(-1,\\text{1}\\right)[\/latex]<\/p>\n<p>27.\u00a0[latex]C\\left(t\\right)=\\frac{8+2t}{300+20t}[\/latex]<\/p>\n<p>28.\u00a0[latex]A\\left(x\\right)=50{x}^{2}+\\frac{800}{x}[\/latex]. 2 by 2 by 5 feet.<\/p>\n<h2>Modeling Using Variation Solutions<\/h2>\n<p>1.\u00a0[latex]y=5{x}^{2}[\/latex]<\/p>\n<p>2.\u00a0[latex]y=10{x}^{3}[\/latex]<\/p>\n<p>3.\u00a0[latex]y=\\frac{18}{{x}^{2}}[\/latex]<\/p>\n<p>4.\u00a0[latex]y=\\frac{20}{\\sqrt[3]{x}}[\/latex]<\/p>\n<p>5.\u00a0[latex]y=10xzw[\/latex]<\/p>\n<p>6.\u00a0[latex]y=10x\\sqrt{z}[\/latex]<\/p>\n<p>7.\u00a0[latex]y=4\\frac{xz}{w}[\/latex]<\/p>\n<p>8.\u00a0[latex]y=256[\/latex]<\/p>\n<p>9.\u00a0[latex]y=6[\/latex]<\/p>\n<p>10.\u00a0[latex]y=18[\/latex]<\/p>\n<p>11.\u00a0[latex]y=\\frac{81}{2}[\/latex]<\/p>\n<p>12.\u00a03 seconds<\/p>\n<p>13.\u00a048 inches<\/p>\n","protected":false},"author":13,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"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\/292"}],"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\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/292\/revisions"}],"predecessor-version":[{"id":395,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/292\/revisions\/395"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/292\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=292"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=292"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=292"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}