{"id":6658,"date":"2011-03-24T23:36:49","date_gmt":"2011-03-24T23:36:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6658"},"modified":"2022-01-13T00:41:56","modified_gmt":"2022-01-13T00:41:56","slug":"mostre-que-fg-e-uma-funcao-racional","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6658","title":{"rendered":"Mostre que $f+g$ \u00e9 uma fun\u00e7\u00e3o racional"},"content":{"rendered":"<p><ul id='GTTabs_ul_6658' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6658' class='GTTabs_curr'><a  id=\"6658_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6658' ><a  id=\"6658_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_6658'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sejam: \\[\\begin{matrix}<br \/>\nf:x\\to \\frac{3x-4}{{{(x-1)}^{2}}} &amp; e &amp; g:x\\to \\frac{4}{{{x}^{3}}-1}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<p>Mostre que $f+g$ \u00e9 uma fun\u00e7\u00e3o racional e determine o seu dom\u00ednio.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6658' onClick='GTTabs_show(1,6658)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6658'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Sejam: \\[\\begin{matrix}<br \/>\nf:x\\to \\frac{3x-4}{{{(x-1)}^{2}}} &amp; e &amp; g:x\\to \\frac{4}{{{x}^{3}}-1}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>${{D}_{f}}=\\left\\{ x\\in \\mathbb{R}:{{(x-1)}^{2}}\\ne 0 \\right\\}=\\mathbb{R}\\backslash \\left\\{ 1 \\right\\}$<\/p>\n<p>${{D}_{g}}=\\left\\{ x\\in \\mathbb{R}:{{x}^{3}}-1\\ne 0 \\right\\}=\\mathbb{R}\\backslash \\left\\{ 1 \\right\\}$<\/p>\n<p>${{D}_{f+g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ 1 \\right\\}$<\/p>\n<p>Ora,<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n(f+g)(x) &amp; = &amp; f(x)+g(x)\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3x-4}{{{(x-1)}^{2}}}+\\frac{4}{{{x}^{3}}-1}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3x-4}{(x-1)(x-1)}+\\frac{4}{(x-1)({{x}^{2}}+x+1)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{(3x-4)({{x}^{2}}+x+1)}{(x-1)(x-1)({{x}^{2}}+x+1)}+\\frac{4(x-1)}{(x-1)(x-1)({{x}^{2}}+x+1)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3{{x}^{3}}+3{{x}^{2}}+3x-4{{x}^{2}}-4x-4+4x-4}{({{x}^{2}}-2x+1)({{x}^{2}}+x+1)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3{{x}^{3}}-{{x}^{2}}+3x-8}{{{x}^{4}}+{{x}^{3}}+{{x}^{2}}-2{{x}^{3}}-2{{x}^{2}}-2x+{{x}^{2}}+x+1}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3{{x}^{3}}-{{x}^{2}}+3x-8}{{{x}^{4}}-{{x}^{3}}-x+1}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p><strong>C\u00e1lculo auxiliar<\/strong>:<\/p>\n<p>$\\begin{matrix}<br \/>\n{} &amp; 1 &amp; 0 &amp; 0 &amp; -1\u00a0 \\\\<br \/>\n1 &amp; {} &amp; 1 &amp; 1 &amp; 1\u00a0 \\\\<br \/>\n{} &amp; 1 &amp; 1 &amp; 1 &amp; 0\u00a0 \\\\<br \/>\n\\end{matrix}$<\/p>\n<p>(Regra de Ruffini para a factoriza\u00e7\u00e3o de ${{x}^{3}}-1$)<\/p>\n<p>Portanto, \\[\\begin{array}{*{35}{l}}<br \/>\nf+g: &amp; \\mathbb{R}\\backslash \\left\\{ 1 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{3{{x}^{3}}-{{x}^{2}}+3x-8}{{{x}^{4}}-{{x}^{3}}-x+1}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>A fun\u00e7\u00e3o $f+g$ \u00e9 racional pois \u00e9 definida por uma express\u00e3o que \u00e9 um quociente de dois polin\u00f3mios.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6658' onClick='GTTabs_show(0,6658)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sejam: \\[\\begin{matrix} f:x\\to \\frac{3x-4}{{{(x-1)}^{2}}} &amp; e &amp; g:x\\to \\frac{4}{{{x}^{3}}-1}\u00a0 \\\\ \\end{matrix}\\] Mostre que $f+g$ \u00e9 uma fun\u00e7\u00e3o racional e determine o seu dom\u00ednio. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19480,"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-6658","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":3072,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/Funcao_fg_P200_Ex56.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6658","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=6658"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6658\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19480"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6658"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6658"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6658"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6658"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}