{"id":6743,"date":"2011-04-11T15:29:20","date_gmt":"2011-04-11T14:29:20","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6743"},"modified":"2022-01-11T23:57:04","modified_gmt":"2022-01-11T23:57:04","slug":"sendo-f-e-g-funcoes-reais-de-variavel-real","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6743","title":{"rendered":"Sendo f e g fun\u00e7\u00f5es reais de vari\u00e1vel real"},"content":{"rendered":"<p><ul id='GTTabs_ul_6743' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6743' class='GTTabs_curr'><a  id=\"6743_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6743' ><a  id=\"6743_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_6743'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sendo f e g fun\u00e7\u00f5es reais de vari\u00e1vel real, caracterize $f\\circ g$ e $g\\circ f$, em cada um dos casos:<\/p>\n<ol>\n<li>$\\begin{matrix}<br \/>\nf(x)=\\sqrt{x} &amp; \\text{e} &amp; g(x)={{x}^{2}}+1\u00a0 \\\\<br \/>\n\\end{matrix}$<\/li>\n<li>$\\begin{matrix}<br \/>\nf(x)={{(x-1)}^{3}} &amp; \\text{e} &amp; g(x)=\\sqrt[3]{x}+1\u00a0 \\\\<br \/>\n\\end{matrix}$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6743' onClick='GTTabs_show(1,6743)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6743'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, ${{D}_{f\\circ g}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{g}}\\wedge g(x)\\in {{D}_{f}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\wedge ({{x}^{2}}+1)\\in \\mathbb{R}_{0}^{+} \\right\\}=\\mathbb{R}$.\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\n(f\\circ g)(x) &amp; = &amp; f(g(x))\u00a0 \\\\<br \/>\n{} &amp; = &amp; f({{x}^{2}}+1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\sqrt{{{x}^{2}}+1}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf\\circ g: &amp; \\mathbb{R}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to \\sqrt{{{x}^{2}}+1}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Ora, ${{D}_{g\\circ f}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{f}}\\wedge f(x)\\in {{D}_{g}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}_{0}^{+}\\wedge (\\sqrt{x})\\in \\mathbb{R} \\right\\}=\\mathbb{R}_{0}^{+}$.<\/p>\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\n(g\\circ f)(x) &amp; = &amp; g(f(x))\u00a0 \\\\<br \/>\n{} &amp; = &amp; g(\\sqrt{x})\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\left| x \\right|+1\u00a0 \\\\<br \/>\n{} &amp; = &amp; x+1\\,\\,(\\text{pois }x\\in \\mathbb{R}_{0}^{+})\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\ng\\circ f: &amp; \\mathbb{R}_{0}^{+}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to x+1\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora, ${{D}_{f\\circ g}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{g}}\\wedge g(x)\\in {{D}_{f}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\wedge (\\sqrt[3]{x}+1)\\in \\mathbb{R} \\right\\}=\\mathbb{R}$.\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\n(f\\circ g)(x) &amp; = &amp; f(g(x))\u00a0 \\\\<br \/>\n{} &amp; = &amp; f(\\sqrt[3]{x}+1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{(\\sqrt[3]{x}+1-1)}^{3}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; x\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\nf\\circ g: &amp; \\mathbb{R}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to x\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Ora, ${{D}_{g\\circ f}}=\\left\\{ x\\in \\mathbb{R}:x\\in {{D}_{f}}\\wedge f(x)\\in {{D}_{g}} \\right\\}=\\left\\{ x\\in \\mathbb{R}:x\\in \\mathbb{R}\\wedge ({{(x-1)}^{3}})\\in \\mathbb{R} \\right\\}=\\mathbb{R}$.<\/p>\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\n(g\\circ f)(x) &amp; = &amp; g(f(x))\u00a0 \\\\<br \/>\n{} &amp; = &amp; g({{(x-1)}^{3}})\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\sqrt[3]{{{(x-1)}^{3}}}+1\u00a0 \\\\<br \/>\n{} &amp; = &amp; x-1+1\u00a0 \\\\<br \/>\n{} &amp; = &amp; x\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o \\[\\begin{array}{*{35}{l}}<br \/>\ng\\circ f: &amp; \\mathbb{R}\\to \\mathbb{R}\u00a0 \\\\<br \/>\n{} &amp; x\\to x\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6743' onClick='GTTabs_show(0,6743)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sendo f e g fun\u00e7\u00f5es reais de vari\u00e1vel real, caracterize $f\\circ g$ e $g\\circ f$, em cada um dos casos: $\\begin{matrix} f(x)=\\sqrt{x} &amp; \\text{e} &amp; g(x)={{x}^{2}}+1\u00a0 \\\\ \\end{matrix}$ $\\begin{matrix} f(x)={{(x-1)}^{3}} &amp;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19175,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,157],"tags":[158],"series":[],"class_list":["post-6743","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-com-radicais","tag-funcoes-com-radicais-2"],"views":2723,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat66.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6743","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=6743"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6743\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6743"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6743"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6743"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}