{"id":11815,"date":"2014-02-24T16:18:50","date_gmt":"2014-02-24T16:18:50","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11815"},"modified":"2022-01-22T02:28:47","modified_gmt":"2022-01-22T02:28:47","slug":"considera-a-funcao-cujo-grafico-esta-representado-na-figura","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11815","title":{"rendered":"Considere a fun\u00e7\u00e3o cujo gr\u00e1fico est\u00e1 representado na figura"},"content":{"rendered":"<p><ul id='GTTabs_ul_11815' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11815' class='GTTabs_curr'><a  id=\"11815_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11815' ><a  id=\"11815_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_11815'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11816\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11816\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\" data-orig-size=\"527,524\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1393258666&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\" class=\"alignright  wp-image-11816\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\" alt=\"Gr\u00e1fico\" width=\"316\" height=\"314\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg 527w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3-150x150.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3-300x298.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3-400x397.jpg 400w\" sizes=\"auto, (max-width: 316px) 100vw, 316px\" \/><\/a>Considere a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\left] { &#8211; \\infty , &#8211; 1} \\right[ \\cup \\left[ {1, + \\infty } \\right[$, cujo gr\u00e1fico est\u00e1 representado na figura.<\/p>\n<p>Determine um express\u00e3o que defina a fun\u00e7\u00e3o.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11815' onClick='GTTabs_show(1,11815)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11815'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>O tro\u00e7o da esquerda \u00e9 um arco de par\u00e1bola, a qual pode ser definida por $y = a\\left( {x + \\frac{7}{4}} \\right)\\left( {x &#8211; {x_2}} \\right)$, pois $ &#8211; \\frac{7}{4}$ \u00e9 um zero da fun\u00e7\u00e3o.<\/p>\n<p>Como os pontos de coordenadas $\\left( { &#8211; 2, &#8211; 1} \\right)$ e $\\left( { &#8211; 1,2} \\right)$ pertencem \u00e0 par\u00e1bola, vem:<\/p>\n<p>\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; 1 = a\\left( { &#8211; 2 + \\frac{7}{4}} \\right)\\left( { &#8211; 2 &#8211; {x_2}} \\right)} \\\\<br \/>\n{2 = a\\left( { &#8211; 1 + \\frac{7}{4}} \\right)\\left( { &#8211; 1 &#8211; {x_2}} \\right)}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; 1 =\u00a0 &#8211; \\frac{1}{4}a\\left( { &#8211; 2 &#8211; {x_2}} \\right)} \\\\<br \/>\n{2 = \\frac{3}{4}a\\left( { &#8211; 1 &#8211; {x_2}} \\right)}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}<br \/>\n{\\left( { &#8211; 1 \\times } \\right)} \\\\<br \/>\n{}<br \/>\n\\end{array}\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{ &#8211; 2a &#8211; a{x_2} = 4} \\\\<br \/>\n{ &#8211; a &#8211; a{x_2} = \\frac{8}{3}}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow\u00a0 \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{a =\u00a0 &#8211; \\frac{4}{3}} \\\\<br \/>\n{\\frac{4}{3} + \\frac{4}{3}{x_2} = \\frac{8}{3}}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{a =\u00a0 &#8211; \\frac{4}{3}} \\\\<br \/>\n{{x_2} = 1}<br \/>\n\\end{array}} \\right.}&amp;{}<br \/>\n\\end{array}\\]<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11816\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11816\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\" data-orig-size=\"527,524\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1393258666&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\" class=\"alignright  wp-image-11816\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg\" alt=\"Gr\u00e1fico\" width=\"316\" height=\"314\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3.jpg 527w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3-150x150.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3-300x298.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11-2-136-3-400x397.jpg 400w\" sizes=\"auto, (max-width: 316px) 100vw, 316px\" \/><\/a>Logo, a par\u00e1bola pode ser definida por:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\ny&amp; = &amp;{ &#8211; \\frac{4}{3}\\left( {x + \\frac{7}{4}} \\right)\\left( {x &#8211; 1} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; \\frac{4}{3}\\left( {{x^2} + \\frac{3}{4}x &#8211; \\frac{7}{4}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; \\frac{4}{3}{x^2} &#8211; x + \\frac{7}{3}}<br \/>\n\\end{array}\\]<\/p>\n<p>As retas que cont\u00eam os tro\u00e7os central e da direita t\u00eam, respetivamente, as seguintes equa\u00e7\u00f5es reduzidas: $y = 2x$ e $y =\u00a0 &#8211; x + 5$.<\/p>\n<p>Assim, temos:<\/p>\n<p>\\[f\\left( x \\right) = \\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n{ &#8211; \\frac{4}{3}{x^2} &#8211; x + \\frac{7}{3}}&amp; \\Leftarrow &amp;{x &lt;\u00a0 &#8211; 1} \\\\<br \/>\n{2x}&amp; \\Leftarrow &amp;{1 \\leqslant x &lt; 3} \\\\<br \/>\n{ &#8211; x + 5}&amp; \\Leftarrow &amp;{x \\geqslant 3}<br \/>\n\\end{array}} \\right.\\]<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11815' onClick='GTTabs_show(0,11815)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere a fun\u00e7\u00e3o $f$, de dom\u00ednio $\\left] { &#8211; \\infty , &#8211; 1} \\right[ \\cup \\left[ {1, + \\infty } \\right[$, cujo gr\u00e1fico est\u00e1 representado na figura. Determine um express\u00e3o que&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20881,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,133],"tags":[422,359],"series":[],"class_list":["post-11815","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-definidas-por-ramos","tag-11-o-ano","tag-funcao-definida-por-ramos"],"views":4112,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/11V2Pag136-3_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11815","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=11815"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11815\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20881"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11815"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11815"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11815"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11815"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}