{"id":6657,"date":"2011-03-24T22:53:17","date_gmt":"2011-03-24T22:53:17","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6657"},"modified":"2022-01-13T00:40:19","modified_gmt":"2022-01-13T00:40:19","slug":"sejam-as-funcoes-racionais","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6657","title":{"rendered":"Sejam as fun\u00e7\u00f5es racionais"},"content":{"rendered":"<p><ul id='GTTabs_ul_6657' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6657' class='GTTabs_curr'><a  id=\"6657_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6657' ><a  id=\"6657_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_6657'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sejam as fun\u00e7\u00f5es racionais definidas por: \\[\\begin{matrix}<br \/>\nf(x)=\\frac{1}{4x+3} &amp; e &amp; g(x)=\\frac{2x-1}{(4x+3)(x-7)}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<ol>\n<li>Indique o seu dom\u00ednio.<\/li>\n<li>Caracterize $f+g$.<\/li>\n<li>Determine $x\\in \\mathbb{R}$ tal que $f(x)\\le g(x)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6657' onClick='GTTabs_show(1,6657)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6657'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Sejam as fun\u00e7\u00f5es racionais definidas por: \\[\\begin{matrix}<br \/>\nf(x)=\\frac{1}{4x+3} &amp; e &amp; g(x)=\\frac{2x-1}{(4x+3)(x-7)}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>${{D}_{f}}=\\left\\{ x\\in \\mathbb{R}:4x+3\\ne 0 \\right\\}=\\mathbb{R}\\backslash \\left\\{ -\\frac{3}{4} \\right\\}$\n<p>${{D}_{g}}=\\left\\{ x\\in \\mathbb{R}:(4x+3)(x-7)\\ne 0 \\right\\}=\\mathbb{R}\\backslash \\left\\{ -\\frac{3}{4},7 \\right\\}$<\/p>\n<\/li>\n<li>${{D}_{f+g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ -\\frac{3}{4},7 \\right\\}$\\[\\begin{array}{*{35}{l}}<br \/>\n(f+g)(x) &amp; = &amp; f(x)+g(x)\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{1}{4x+3}+\\frac{2x-1}{(4x+3)(x-7)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{(x-7)+2x-1}{(4x+3)(x-7)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3x-8}{4{{x}^{2}}-25x-21}\u00a0 \\\\<br \/>\n\\end{array}\\]Logo,<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\nf+g: &amp; \\mathbb{R}\\backslash \\left\\{ -\\frac{3}{4},7 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{3x-8}{4{{x}^{2}}-25x-21}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<\/li>\n<li>Ora, \\[\\begin{array}{*{35}{l}}<br \/>\nf(x)\\le g(x) &amp; \\Leftrightarrow\u00a0 &amp; \\frac{1}{4x+3}\\le \\frac{2x-1}{(4x+3)(x-7)}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{(x-7)-(2x-1)}{(4x+3)(x-7)}\\le 0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{-x-6}{(4x+3)(x-7)}\\le 0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{x+6}{(4x+3)(x-7)}\\ge 0\u00a0 \\\\<br \/>\n\\end{array}\\]Assim, vem:<\/p>\n<table border=\"2\" cellspacing=\"4\" cellpadding=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$x$<\/td>\n<td style=\"text-align: left;\">$-\\infty $<\/td>\n<td style=\"text-align: center;\">$-6$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$-\\frac{3}{4}$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$7$<\/td>\n<td style=\"text-align: right;\">\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 $+\\infty $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$x+6$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$(4x+3)(x-7)$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$\\frac{x+6}{(4x+3)(x-7)}$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">n.d.<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Portanto, $f(x)\\le g(x)\\Leftrightarrow x\\in \\left[ -6,-\\frac{3}{4} \\right[\\cup \\left] 7,+\\infty\u00a0 \\right[$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6657' onClick='GTTabs_show(0,6657)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sejam as fun\u00e7\u00f5es racionais definidas por: \\[\\begin{matrix} f(x)=\\frac{1}{4x+3} &amp; e &amp; g(x)=\\frac{2x-1}{(4x+3)(x-7)}\u00a0 \\\\ \\end{matrix}\\] Indique o seu dom\u00ednio. Caracterize $f+g$. Determine $x\\in \\mathbb{R}$ tal que $f(x)\\le g(x)$. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19478,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,147],"tags":[148],"series":[],"class_list":["post-6657","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-operacoes-com-funcoes","tag-operacoes-com-funcoes-2"],"views":1948,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/Funcao_f-P200_Ex55.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6657","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6657"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6657\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19478"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6657"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6657"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6657"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}