{"id":7346,"date":"2012-01-20T12:00:22","date_gmt":"2012-01-20T12:00:22","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7346"},"modified":"2022-01-14T00:13:30","modified_gmt":"2022-01-14T00:13:30","slug":"considere-as-funcoes-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7346","title":{"rendered":"Considere as fun\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_7346' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7346' class='GTTabs_curr'><a  id=\"7346_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7346' ><a  id=\"7346_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_7346'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere as fun\u00e7\u00f5es<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\nf:x\\to \\frac{4-\\ln (2-x)}{3}\u00a0 \\\\<br \/>\ng:x\\to 2+3{{e}^{2x-1}}\u00a0 \\\\<br \/>\nh:x\\to {{\\log }_{2}}(2x-2)-{{\\log }_{2}}(x+2)-2\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<ol>\n<li>Indique o dom\u00ednio de cada uma das fun\u00e7\u00f5es.<\/li>\n<li>Caraterize as fun\u00e7\u00f5es inversas de $f$ e $g$.<\/li>\n<li>Determine os zeros de cada uma das fun\u00e7\u00f5es.<\/li>\n<li>Determine os valores de $x$ para os quais $h(x)\\le -2$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7346' onClick='GTTabs_show(1,7346)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7346'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Considere as fun\u00e7\u00f5es<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\nf:x\\to \\frac{4-\\ln (2-x)}{3}\u00a0 \\\\<br \/>\ng:x\\to 2+3{{e}^{2x-1}}\u00a0 \\\\<br \/>\nh:x\\to {{\\log }_{2}}(2x-2)-{{\\log }_{2}}(x+2)-2\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Os dom\u00ednios das fun\u00e7\u00f5es s\u00e3o:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n{{D}_{f}} &amp; = &amp; \\left\\{ x\\in \\mathbb{R}:2-x&gt;0 \\right\\}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\left\\{ x\\in \\mathbb{R}:x&lt;2 \\right\\}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\left] -\\infty ,2 \\right[\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n{{D}_{g}} &amp; = &amp; \\left\\{ x\\in \\mathbb{R}:(2x-1)\\in \\mathbb{R} \\right\\}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\mathbb{R}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n{{D}_{h}} &amp; = &amp; \\left\\{ x\\in \\mathbb{R}:2x-2&gt;0\\wedge x+2&gt;0 \\right\\}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\left\\{ x\\in \\mathbb{R}:x&gt;1\\wedge x&gt;-2 \\right\\}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\left] 1,+\\infty\u00a0 \\right[\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>Ora, \\[\\begin{array}{*{35}{l}}<br \/>\ny=\\frac{4-\\ln (2-x)}{3} &amp; \\Leftrightarrow\u00a0 &amp; \\ln (2-x)=4-3y\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 2-x={{e}^{4-3y}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x=2-{{e}^{-3y+4}}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nAssim, \\[\\begin{array}{*{20}{l}}{{{D&#8217;}_f}}&amp; = &amp;{\\left\\{ {x \\in \\mathbb{R}:( &#8211; 3x + 4) \\in \\mathbb{R}} \\right\\}{\\rm{\\;}}}\\\\{}&amp; = &amp;{\\mathbb{R}{\\rm{\\;}}}\\end{array}\\]<br \/>\nLogo, a fun\u00e7\u00e3o inversa de $f$ pode ser assim caraterizada: \\[\\begin{array}{*{35}{l}}<br \/>\n{{f}^{-1}}: &amp; \\mathbb{R}\\to \\left] -\\infty ,2 \\right[\u00a0 \\\\<br \/>\n{} &amp; x\\to 2-{{e}^{-3x+4}}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Ora, \\[\\begin{array}{*{35}{l}}<br \/>\ny=2+3{{e}^{2x-1}} &amp; \\Leftrightarrow\u00a0 &amp; {{e}^{2x-1}}=\\frac{y-2}{3}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 2x-1=\\ln \\left( \\frac{y-2}{3} \\right)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x=\\frac{1}{2}+\\frac{1}{2}\\ln \\left( \\frac{y-2}{3} \\right)\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nAssim, \\[\\begin{array}{*{20}{l}}{{{D&#8217;}_g}}&amp; = &amp;{\\left\\{ {x \\in \\mathbb{R}:\\frac{{x &#8211; 2}}{3} &gt; 0} \\right\\}{\\rm{\\;}}}\\\\{}&amp; = &amp;{\\left] {2, + \\infty {\\rm{\\;}}} \\right[{\\rm{\\;}}}\\end{array}\\]<br \/>\nLogo, a fun\u00e7\u00e3o inversa de $g$ pode ser assim caraterizada: \\[\\begin{array}{*{35}{l}}<br \/>\n{{g}^{-1}}: &amp; \\left] 2,+\\infty\u00a0 \\right[\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{1}{2}+\\frac{1}{2}\\ln \\left( \\frac{x-2}{3} \\right)\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Como \\[\\begin{array}{*{35}{l}}<br \/>\nf(x)=0 &amp; \\Leftrightarrow\u00a0 &amp; \\frac{4-\\ln (2-x)}{3}=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\ln (2-x)=4\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 2-x={{e}^{4}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x=2-{{e}^{4}}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\na fun\u00e7\u00e3o $f$ apenas possui um zero: $x=2-{{e}^{4}}$.<\/p>\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\ng(x)=0 &amp; \\Leftrightarrow\u00a0 &amp; 2+3{{e}^{2x-1}}=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x\\in \\left\\{ {} \\right\\},\\text{ pois }{{e}^{2x-1}}&gt;0,\\forall x\\in \\mathbb{R}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\na fun\u00e7\u00e3o $g$ n\u00e3o tem zeros.<\/p>\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\nh(x)=0 &amp; \\Leftrightarrow\u00a0 &amp; {{\\log }_{2}}(2x-2)-{{\\log }_{2}}(x+2)-2=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\log }_{2}}(2x-2)-{{\\log }_{2}}4={{\\log }_{2}}(x+2)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{\\log }_{2}}\\left( \\frac{2x-2}{4} \\right)={{\\log }_{2}}(x+2)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix}<br \/>\n\\frac{2x-2}{4}=x+2 &amp; \\wedge\u00a0 &amp; x\\in {{D}_{h}}\u00a0 \\\\<br \/>\n\\end{matrix}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix}<br \/>\n2x-2=4x+8 &amp; \\wedge\u00a0 &amp; x\\in \\left] 1,+\\infty\u00a0 \\right[\u00a0 \\\\<br \/>\n\\end{matrix}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{matrix}<br \/>\nx=-5 &amp; \\wedge\u00a0 &amp; x\\in \\left] 1,+\\infty\u00a0 \\right[\u00a0 \\\\<br \/>\n\\end{matrix}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x\\in \\left\\{ {} \\right\\}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\na fun\u00e7\u00e3o $h$ n\u00e3o tem zeros.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora,<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{h(x) \\leqslant\u00a0 &#8211; 2}&amp; \\Leftrightarrow &amp;{{{\\log }_2}(2x &#8211; 2) &#8211; {{\\log }_2}(x + 2) &#8211; 2 \\leqslant\u00a0 &#8211; 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{{\\log }_2}(2x &#8211; 2) \\leqslant {{\\log }_2}(x + 2)} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{2x &#8211; 2 \\leqslant x + 2}&amp; \\wedge &amp;{x \\in {D_h},{\\text{ pois }}x \\to {{\\log }_2}x{\\text{ }}{\\text{ \u00e9 estritamente crescente}}}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x \\leqslant 4}&amp; \\wedge &amp;{x \\in \\left] {1, + \\infty } \\right[}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left] {1,4} \\right]}<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7346' onClick='GTTabs_show(0,7346)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere as fun\u00e7\u00f5es $$\\begin{array}{*{35}{l}} f:x\\to \\frac{4-\\ln (2-x)}{3}\u00a0 \\\\ g:x\\to 2+3{{e}^{2x-1}}\u00a0 \\\\ h:x\\to {{\\log }_{2}}(2x-2)-{{\\log }_{2}}(x+2)-2\u00a0 \\\\ \\end{array}$$ Indique o dom\u00ednio de cada uma das fun\u00e7\u00f5es. Caraterize as fun\u00e7\u00f5es inversas de $f$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19177,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,267],"tags":[427,268,156,275],"series":[],"class_list":["post-7346","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-exponenciais-e-logaritmicas","tag-12-o-ano","tag-funcao-exponencial","tag-funcao-inversa-2","tag-funcao-logaritmica"],"views":1929,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat68.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7346","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=7346"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7346\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19177"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7346"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}