{"id":6660,"date":"2011-03-25T01:38:51","date_gmt":"2011-03-25T01:38:51","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6660"},"modified":"2022-01-13T00:44:23","modified_gmt":"2022-01-13T00:44:23","slug":"f-e-uma-funcao-racional","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6660","title":{"rendered":"f \u00e9 uma fun\u00e7\u00e3o racional"},"content":{"rendered":"<p><ul id='GTTabs_ul_6660' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6660' class='GTTabs_curr'><a  id=\"6660_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6660' ><a  id=\"6660_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_6660'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>f \u00e9 uma fun\u00e7\u00e3o racional definida em $\\mathbb{R}\\backslash \\left\\{ 1 \\right\\}$ por \\[f(x)=\\frac{-2{{x}^{2}}+6x-3}{2{{(x-1)}^{2}}}\\]<\/p>\n<p>Encontre os reais a, b e c tais que, para todo o $x\\ne 1$, \\[f(x)=a+\\frac{b}{x-1}+\\frac{c}{2{{(x-1)}^{2}}}\\]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6660' onClick='GTTabs_show(1,6660)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6660'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>f \u00e9 uma fun\u00e7\u00e3o racional definida em $\\mathbb{R}\\backslash \\left\\{ 1 \\right\\}$ por \\[f(x)=\\frac{-2{{x}^{2}}+6x-3}{2{{(x-1)}^{2}}}\\]<\/p>\n<p>Encontre os reais a, b e c tais que, para todo o $x\\ne 1$, \\[f(x)=a+\\frac{b}{x-1}+\\frac{c}{2{{(x-1)}^{2}}}\\]<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p>Ora, \\[\\begin{array}{*{35}{l}}<br \/>\nf(x) &amp; = &amp; a+\\frac{b}{x-1}+\\frac{c}{2{{(x-1)}^{2}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2a{{(x-1)}^{2}}}{2{{(x-1)}^{2}}}+\\frac{2b(x-1)}{2{{(x-1)}^{2}}}+\\frac{c}{2{{(x-1)}^{2}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2a{{x}^{2}}-4ax+2a+2bx-2b+c}{2{{(x-1)}^{2}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2a{{x}^{2}}+(-4a+2b)x+(2a-2b+c)}{2{{(x-1)}^{2}}}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Dado que \\[f(x)=\\frac{-2{{x}^{2}}+6x-3}{2{{(x-1)}^{2}}}\\] vem:<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n2a=-2\u00a0 \\\\<br \/>\n-4a+2b=6\u00a0 \\\\<br \/>\n2a-2b+c=-3\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=-1\u00a0 \\\\<br \/>\n2b=2\u00a0 \\\\<br \/>\n-2b+c=-1\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=-1\u00a0 \\\\<br \/>\nb=1\u00a0 \\\\<br \/>\nc=1\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6660' onClick='GTTabs_show(0,6660)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado f \u00e9 uma fun\u00e7\u00e3o racional definida em $\\mathbb{R}\\backslash \\left\\{ 1 \\right\\}$ por \\[f(x)=\\frac{-2{{x}^{2}}+6x-3}{2{{(x-1)}^{2}}}\\] Encontre os reais a, b e c tais que, para todo o $x\\ne 1$, \\[f(x)=a+\\frac{b}{x-1}+\\frac{c}{2{{(x-1)}^{2}}}\\] Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19482,"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-6660","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":1792,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/Funcao_f-P200_Ex58.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6660","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=6660"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6660\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19482"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6660"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6660"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}