{"id":24969,"date":"2023-04-11T12:33:07","date_gmt":"2023-04-11T11:33:07","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=24969"},"modified":"2023-04-11T15:46:06","modified_gmt":"2023-04-11T14:46:06","slug":"cinco-funcoes-afins","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=24969","title":{"rendered":"Cinco fun\u00e7\u00f5es afins"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_24969' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_24969' class='GTTabs_curr'><a  id=\"24969_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_24969' ><a  id=\"24969_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_24969'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considera as fun\u00e7\u00f5es, definidas em \\(\\mathbb{Q}\\), por:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 20%;\">\\(f\\left( x \\right) = &#8211; 4x\\)<\/td>\n<td style=\"width: 20%;\">\\(g\\left( x \\right) = &#8211; 3\\)<\/td>\n<td style=\"width: 20%;\">\\(h\\left( x \\right) = x + 2\\)<\/td>\n<td style=\"width: 20%;\">\\(i\\left( x \\right) = &#8211; x &#8211; \\frac{2}{3}\\)<\/td>\n<td style=\"width: 20%;\">\\(j\\left( x \\right) = \\frac{x}{4}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Quais das fun\u00e7\u00f5es s\u00e3o fun\u00e7\u00f5es constantes? E quais s\u00e3o fun\u00e7\u00f5es lineares? E quais s\u00e3o fun\u00e7\u00f5es afins?<\/li>\n<li>Indica o coeficiente de cada fun\u00e7\u00e3o linear.<\/li>\n<li>Considerando a forma can\u00f3nica de cada uma das fun\u00e7\u00f5es afins que n\u00e3o s\u00e3o constantes, nem lineares, indica, para cada caso, o coeficiente de <em>x<\/em> e o termo independente.<\/li>\n<li>Qual \u00e9 a imagem de \\( &#8211; \\frac{7}{5}\\) por cada uma das fun\u00e7\u00f5es?<\/li>\n<li>Qual \u00e9 o objeto cuja imagem \u00e9 12 pela fun\u00e7\u00e3o \\(j\\)? Mostra como chegaste \u00e0 tua resposta.<\/li>\n<li>Calcula:<br \/>a) \\(\\left( {f + g} \\right)\\left( 3 \\right)\\)<br \/>b) \\(\\left( {f \\times h} \\right)\\left( { &#8211; 2} \\right)\\)<\/li>\n<li>Escreve a forma can\u00f3nica da fun\u00e7\u00e3o \\(j &#8211; i\\).<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_24969' onClick='GTTabs_show(1,24969)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_24969'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><\/p>\n<blockquote>\n<p>Considera as fun\u00e7\u00f5es, definidas em \\(\\mathbb{Q}\\), por:<\/p>\n<\/blockquote>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 20%;\">\\(f\\left( x \\right) = &#8211; 4x\\)<\/td>\n<td style=\"width: 20%;\">\\(g\\left( x \\right) = &#8211; 3\\)<\/td>\n<td style=\"width: 20%;\">\\(h\\left( x \\right) = x + 2\\)<\/td>\n<td style=\"width: 20%;\">\\(i\\left( x \\right) = &#8211; x &#8211; \\frac{2}{3}\\)<\/td>\n<td style=\"width: 20%;\">\\(j\\left( x \\right) = \\frac{x}{4}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>S\u00e3o fun\u00e7\u00f5es constantes: \\(g\\).<br \/>S\u00e3o fun\u00e7\u00f5es lineares: \\(f\\) e \\(j\\).<br \/>S\u00e3o fun\u00e7\u00f5es afins: \\(f\\), \\(g\\), \\(h\\), \\(i\\) e \\(j\\).<br \/><br \/><\/li>\n<li>O coeficiente da fun\u00e7\u00e3o linear \\(f\\) \u00e9 \\( &#8211; 4\\).<br \/>O coeficiente da fun\u00e7\u00e3o linear \\(j\\) \u00e9 \\(\\frac{1}{4}\\).<br \/><br \/><\/li>\n<li>As fun\u00e7\u00f5es afins que n\u00e3o s\u00e3o constantes, nem lineares, s\u00e3o \\(h\\) e \\(i\\).<br \/>Relativamente a \\(h\\), o coeficiente de <em>x<\/em> \u00e9 \\(1\\) e o termo independente \u00e9 \\(2\\).<br \/>Relativamente a \\(i\\), o coeficiente de <em>x<\/em> \u00e9 \\( &#8211; 1\\) e o termo independente \u00e9 \\( &#8211; \\frac{2}{3}\\).<br \/><br \/><\/li>\n<li>Seguidamente, apresenta-se a imagem de \\( &#8211; \\frac{7}{5}\\) por cada uma das fun\u00e7\u00f5es:<br \/>\\[\\begin{array}{l}f\\left( { &#8211; \\frac{7}{5}} \\right) = &#8211; 4 \\times \\left( { &#8211; \\frac{7}{5}} \\right) = \\frac{{28}}{5}\\\\g\\left( { &#8211; \\frac{7}{5}} \\right) = &#8211; 3\\\\h\\left( { &#8211; \\frac{7}{5}} \\right) = &#8211; \\frac{7}{5} + 2 = &#8211; \\frac{7}{5} + \\frac{{10}}{5} = \\frac{3}{5}\\\\i\\left( { &#8211; \\frac{7}{5}} \\right) = &#8211; \\left( { &#8211; \\frac{7}{5}} \\right) &#8211; \\frac{2}{3} = \\frac{{21}}{{15}} &#8211; \\frac{{10}}{{15}} = \\frac{{11}}{{15}}\\\\j\\left( { &#8211; \\frac{7}{5}} \\right) = \\frac{{ &#8211; \\frac{7}{5}}}{4} = &#8211; \\frac{7}{5} \\times \\frac{1}{4} = &#8211; \\frac{7}{{20}}\\end{array}\\]<\/li>\n<li>\u00c9 \\(48\\) o objeto cuja imagem \u00e9 12 pela fun\u00e7\u00e3o \\(j\\):<br \/>\\[\\begin{array}{*{20}{l}}{j\\left( x \\right) = 12}&amp; \\Leftrightarrow &amp;{\\frac{x}{4} = 12}\\\\{}&amp; \\Leftrightarrow &amp;{x = 48}\\end{array}\\]<\/li>\n<li>Calculando, vem:<br \/>a) \\(\\left( {f + g} \\right)\\left( 3 \\right) = f\\left( 3 \\right) + g\\left( 3 \\right) = \\left( { &#8211; 4 \\times 3} \\right) + \\left( { &#8211; 3} \\right) = &#8211; 15\\)<br \/>b) \\(\\left( {f \\times h} \\right)\\left( { &#8211; 2} \\right) = f\\left( { &#8211; 2} \\right) \\times h\\left( { &#8211; 2} \\right) = \\left( { &#8211; 4 \\times \\left( { &#8211; 2} \\right)} \\right) \\times \\left( { &#8211; 2 + 2} \\right) = 0\\)<br \/><br \/><\/li>\n<li>A forma can\u00f3nica da fun\u00e7\u00e3o \\(j &#8211; i\\) \u00e9: \\(\\left( {j &#8211; i} \\right)\\left( x \\right) = \\frac{5}{4}x + \\frac{2}{3}\\).<br \/>\\[\\begin{array}{*{20}{l}}{\\left( {j &#8211; i} \\right)\\left( x \\right)}&amp; = &amp;{j\\left( x \\right) &#8211; i\\left( x \\right)}\\\\{}&amp; = &amp;{\\frac{x}{4} &#8211; \\left( { &#8211; x &#8211; \\frac{2}{3}} \\right)}\\\\{}&amp; = &amp;{\\frac{5}{4}x + \\frac{2}{3}}\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_24969' onClick='GTTabs_show(0,24969)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considera as fun\u00e7\u00f5es, definidas em \\(\\mathbb{Q}\\), por: \\(f\\left( x \\right) = &#8211; 4x\\) \\(g\\left( x \\right) = &#8211; 3\\) \\(h\\left( x \\right) = x + 2\\) \\(i\\left( x \\right) = &#8211;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19173,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,709],"tags":[424,345],"series":[],"class_list":["post-24969","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-graficos-de-funcoes-afins","tag-8-o-ano","tag-funcao-afim"],"views":159,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat64.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/24969","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=24969"}],"version-history":[{"count":1,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/24969\/revisions"}],"predecessor-version":[{"id":24988,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/24969\/revisions\/24988"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=24969"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=24969"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=24969"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=24969"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}