{"id":6697,"date":"2011-04-05T15:28:49","date_gmt":"2011-04-05T14:28:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6697"},"modified":"2021-12-26T16:07:01","modified_gmt":"2021-12-26T16:07:01","slug":"dadas-as-funcoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6697","title":{"rendered":"Dadas as fun\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_6697' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6697' class='GTTabs_curr'><a  id=\"6697_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6697' ><a  id=\"6697_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_6697'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Dadas as fun\u00e7\u00f5es definidas em $\\mathbb{R}$ por \\[\\begin{matrix}<br \/>\nf(x)=3x-4 &amp; e &amp; g(x)=\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<ol>\n<li>Determine:<br \/>\n<table style=\"width: 70%;\" border=\"2\" cellspacing=\"4\" cellpadding=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$(f+g)(5)$<\/td>\n<td style=\"text-align: center;\">$(f-g)(5)$<\/td>\n<td style=\"text-align: center;\">$(f\\times g)(5)$<\/td>\n<td style=\"text-align: center;\">$(f\\div g)(5)$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$(f\\circ g)(5)$<\/td>\n<td style=\"text-align: center;\">$(g\\circ f)(5)$<\/td>\n<td style=\"text-align: center;\">$(f\\circ f)(5)$<\/td>\n<td style=\"text-align: center;\">$(g\\circ g)(5)$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Caracterize as fun\u00e7\u00f5es:<br \/>\n<table style=\"width: 70%;\" border=\"2\" cellspacing=\"4\" cellpadding=\"4\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">$f+g$<\/td>\n<td style=\"text-align: center;\">$f-g$<\/td>\n<td style=\"text-align: center;\">$f\\times g$<\/td>\n<td style=\"text-align: center;\">$f\\div g$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$f\\circ g$<\/td>\n<td style=\"text-align: center;\">$g\\circ f$<\/td>\n<td style=\"text-align: center;\">$f\\circ f$<\/td>\n<td style=\"text-align: center;\">$g\\circ g$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6697' onClick='GTTabs_show(1,6697)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6697'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Tem-se sucessivamente:<br \/>\n\\[(f+g)(5)=f(5)+g(5)=11+\\frac{1}{5}=\\frac{56}{5}\\]<br \/>\n\\[(f-g)(5)=f(5)-g(5)=11-\\frac{1}{5}=\\frac{54}{5}\\]<br \/>\n\\[(f\\times g)(5)=f(5)\\times g(5)=11\\times \\frac{1}{5}=\\frac{11}{5}\\]<br \/>\n\\[(f\\div g)(5)=f(5)\\div g(5)=11\\div \\frac{1}{5}=55\\]<br \/>\n\\[(f\\circ g)(5)=f(g(5))=f(\\frac{1}{5})=3\\times \\frac{1}{5}-4=-\\frac{17}{5}\\]<br \/>\n\\[(g\\circ f)(5)=g(f(5))=g(11)=\\frac{1}{11}\\]<br \/>\n\\[(f\\circ f)(5)=f(f(5))=f(11)=29\\]<br \/>\n\\[(g\\circ g)(5)=g(g(5))=g(\\frac{1}{5})=5\\]<\/li>\n<li>Ora, ${{D}_{f}}=\\mathbb{R}$ e ${{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nLogo, ${{D}_{f+g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nComo $(f+g)(x)=f(x)+g(x)=3x-4+\\frac{1}{x}$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf+g: &amp; \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to 3x-4+\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{f-g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nComo $(f-g)(x)=f(x)-g(x)=3x-4-\\frac{1}{x}$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf-g: &amp; \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to 3x-4-\\frac{1}{x}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{f\\times g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nComo $(f\\times g)(x)=f(x)\\times g(x)=\\frac{3x-4}{x}$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf\\times g: &amp; \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{3x-4}{x}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{f\\div g}}={{D}_{f}}\\cap {{D}_{g}}\\cap \\left\\{ x\\in \\mathbb{R}:g(x)\\ne 0 \\right\\}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nComo $(f\\div g)(x)=f(x)\\div g(x)=3{{x}^{2}}-4x$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf\\div g: &amp; \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to 3{{x}^{2}}-4x\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{f\\circ g}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{g}}\\wedge g(x)\\in {{D}_{f}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\wedge (\\frac{1}{x})\\in \\mathbb{R} \\right\\}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nComo $(f\\circ g)(x)=f(g(x))=f(\\frac{1}{x})=\\frac{3}{x}-4$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf\\circ g: &amp; \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{3}{x}-4\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{g\\circ f}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{f}}\\wedge f(x)\\in {{D}_{g}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\wedge (3x-4)\\in \\mathbb{R}\\backslash \\left\\{ 0 \\right\\} \\right\\}=\\mathbb{R}\\backslash \\left\\{ \\frac{4}{3} \\right\\}$.<br \/>\nComo $(g\\circ f)(x)=g(f(x))=g(3x-4)=\\frac{1}{3x-4}$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\ng\\circ f: &amp; \\mathbb{R}\\backslash \\left\\{ \\frac{4}{3} \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\frac{1}{3x-4}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{f\\circ f}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{f}}\\wedge f(x)\\in {{D}_{f}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\wedge (3x-4)\\in \\mathbb{R} \\right\\}=\\mathbb{R}$.<br \/>\nComo $(f\\circ f)(x)=f(f(x))=f(3x-4)=3(3x-4)-4=9x-16$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf\\circ f: &amp; \\mathbb{R}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to 9x-16\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nOra, ${{D}_{g\\circ g}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{g}}\\wedge g(x)\\in {{D}_{g}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\wedge (\\frac{1}{x})\\in \\mathbb{R}\\backslash \\left\\{ 0 \\right\\} \\right\\}=\\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<br \/>\nComo $(g\\circ g)(x)=g(g(x))=g(\\frac{1}{x})=x$, ent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\ng\\circ g: &amp; \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to x\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6697' onClick='GTTabs_show(0,6697)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Dadas as fun\u00e7\u00f5es definidas em $\\mathbb{R}$ por \\[\\begin{matrix} f(x)=3x-4 &amp; e &amp; g(x)=\\frac{1}{x}\u00a0 \\\\ \\end{matrix}\\] Determine: $(f+g)(5)$ $(f-g)(5)$ $(f\\times g)(5)$ $(f\\div g)(5)$ $(f\\circ g)(5)$ $(g\\circ f)(5)$ $(f\\circ f)(5)$ $(g\\circ g)(5)$ Caracterize&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14080,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,153],"tags":[154],"series":[],"class_list":["post-6697","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcao-composta","tag-funcao-composta-2"],"views":2189,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat25.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6697","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=6697"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6697\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14080"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6697"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6697"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6697"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6697"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}