{"id":6659,"date":"2011-03-25T00:14:46","date_gmt":"2011-03-25T00:14:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6659"},"modified":"2022-01-13T00:37:43","modified_gmt":"2022-01-13T00:37:43","slug":"sejam-as-funcoes-f-e-g","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6659","title":{"rendered":"Sejam as fun\u00e7\u00f5es $f$ e $g$"},"content":{"rendered":"<p><ul id='GTTabs_ul_6659' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6659' class='GTTabs_curr'><a  id=\"6659_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6659' ><a  id=\"6659_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_6659'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sejam \\[\\begin{matrix}<br \/>\nf:x\\to \\frac{2x+2}{{{x}^{2}}-3x+2} &amp; e &amp; g:x\\to \\frac{4x-4}{x-2}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<ol>\n<li>Mostre que $f\\times g$ e $\\frac{f}{g}$ s\u00e3o fun\u00e7\u00f5es racionais e determine o seu dom\u00ednio.<\/li>\n<li>Determine os valores de x tais que $f(x)\\le \\frac{1}{2}$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6659' onClick='GTTabs_show(1,6659)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6659'>\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{2x+2}{{{x}^{2}}-3x+2} &amp; e &amp; g:x\\to \\frac{4x-4}{x-2}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>${{D}_{f}}=\\left\\{ x\\in \\mathbb{R}:{{x}^{2}}-3x+2\\ne 0 \\right\\}=\\left\\{ x\\in \\mathbb{R}:\\tilde{\\ }\\left( x=\\frac{3\\pm \\sqrt{9-8}}{2} \\right) \\right\\}=\\mathbb{R}\\backslash \\left\\{ 1,2 \\right\\}$\n<p>${{D}_{g}}=\\left\\{ x\\in \\mathbb{R}:x-2\\ne 0 \\right\\}=\\mathbb{R}\\backslash \\left\\{ 2 \\right\\}$<\/p>\n<p>${{D}_{f\\times g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ 1,2 \\right\\}$<\/p>\n<p>${{D}_{\\frac{f}{g}}}={{D}_{f}}\\cap {{D}_{g}}\\cap \\left\\{ x\\in \\mathbb{R}:g(x)\\ne 0 \\right\\}={{D}_{f}}\\cap {{D}_{g}}\\cap \\left\\{ x\\in \\mathbb{R}:x\\ne 1 \\right\\}=\\mathbb{R}\\backslash \\left\\{ 1,2 \\right\\}$<\/p>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n(f\\times g)(x) &amp; = &amp; f(x)\\times g(x)\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2x+2}{{{x}^{2}}-3x+2}\\times \\frac{4x-4}{x-2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2(x+1)}{(x-1)(x-2)}\\times \\frac{4(x-1)}{x-2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2(x+1)}{(x-2)}\\times \\frac{4}{x-2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{8x+8}{{{x}^{2}}-4x+4}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Logo,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nf\\times g: &amp; \\mathbb{R}\\backslash \\left\\{ 1,2 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{8x+8}{{{x}^{2}}-4x+4}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n(\\frac{f}{g})(x) &amp; = &amp; \\frac{f(x)}{g(x)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2x+2}{{{x}^{2}}-3x+2}\\div \\frac{4x-4}{x-2}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2(x+1)}{(x-1)(x-2)}\\times \\frac{x-2}{4(x-1)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{(x+1)}{(x-1)}\\times \\frac{1}{2(x-1)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{x+1}{2{{x}^{2}}-4x+2}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Logo,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\frac{f}{g}: &amp; \\mathbb{R}\\backslash \\left\\{ 1,2 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{x+1}{2{{x}^{2}}-4x+2}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nf(x)\\le \\frac{1}{2} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{2x+2}{{{x}^{2}}-3x+2}\\le \\frac{1}{2}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{2x+2}{(x-1)(x-2)}-\\frac{1}{2}\\le 0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{4x+4-{{x}^{2}}+3x-2}{2(x-1)(x-2)}\\le 0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\frac{-{{x}^{2}}+7x+2}{2(x-1)(x-2)}\\le 0\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p><strong>C\u00e1lculo auxiliar<\/strong>:<br \/>\n\\[ &#8211; {x^2} + 7x + 2 = 0 \\Leftrightarrow x = \\frac{{ &#8211; 7 \\pm \\sqrt {49 + 8} }}{{ &#8211; 2}} \\Leftrightarrow x = \\frac{{ &#8211; 7 \\pm \\sqrt {57} }}{{ &#8211; 2}} \\Leftrightarrow x = \\frac{{7 &#8211; \\sqrt {57} }}{2} \\vee x = \\frac{{7 + \\sqrt {57} }}{2}\\]<\/p>\n<p>Logo, 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 $<\/td>\n<td style=\"text-align: center;\">$\\frac{7-\\sqrt{57}}{2}$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$1$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$2$<\/td>\n<td><\/td>\n<td style=\"text-align: center;\">$\\frac{7+\\sqrt{57}}{2}$<\/td>\n<td style=\"text-align: right;\">\u00a0\u00a0\u00a0 $+\\infty $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$-{{x}^{2}}+7x+2$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0 +<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0\u00a0 +<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">\u00a0\u00a0 \u00a0+<\/td>\n<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$2(x-1)(x-2)$<\/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<td style=\"text-align: center;\">+<\/td>\n<td style=\"text-align: center;\">+<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$\\frac{-{{x}^{2}}+7x+2}{2(x-1)(x-2)}$<\/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<td style=\"text-align: center;\">$0$<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Portanto, \\[f(x)\\le \\frac{1}{2}\\Leftrightarrow x\\in \\left] -\\infty ,\\frac{7-\\sqrt{57}}{2} \\right]\\cup \\left] 1,2 \\right[\\cup \\left[ \\frac{7+\\sqrt{57}}{2},+\\infty\u00a0 \\right[\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6659' onClick='GTTabs_show(0,6659)'>&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{2x+2}{{{x}^{2}}-3x+2} &amp; e &amp; g:x\\to \\frac{4x-4}{x-2}\u00a0 \\\\ \\end{matrix}\\] Mostre que $f\\times g$ e $\\frac{f}{g}$ s\u00e3o fun\u00e7\u00f5es racionais e determine o seu dom\u00ednio. Determine os valores de x tais&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19481,"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-6659","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":1409,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/Funcao_f-P200_Ex57.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6659","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=6659"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6659\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19481"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6659"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6659"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6659"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}