{"id":11808,"date":"2014-02-24T10:24:36","date_gmt":"2014-02-24T10:24:36","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11808"},"modified":"2022-01-13T01:05:45","modified_gmt":"2022-01-13T01:05:45","slug":"considere-as-funcoes-4","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11808","title":{"rendered":"Considere as fun\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_11808' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11808' class='GTTabs_curr'><a  id=\"11808_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11808' ><a  id=\"11808_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_11808'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere as fun\u00e7\u00f5es definidas por:<\/p>\n<p>\\[\\begin{array}{*{20}{r}}<br \/>\n{\\begin{array}{*{20}{l}}<br \/>\n{f:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to {x^2}}<br \/>\n\\end{array}}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{g:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{1}{{x + 1}}}<br \/>\n\\end{array}}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{h:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to {x^2} &#8211; x}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<\/p>\n<p>Caracterize as seguintes fun\u00e7\u00f5es:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{f + g}&amp;{}&amp;{f \\times g}&amp;{}&amp;{\\frac{f}{g}}&amp;{}&amp;{h &#8211; g}&amp;{}&amp;{\\frac{f}{h}}<br \/>\n\\end{array}\\]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11808' onClick='GTTabs_show(1,11808)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11808'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>\\[\\begin{array}{*{20}{r}}<br \/>\n{\\begin{array}{*{20}{l}}<br \/>\n{f:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to {x^2}}<br \/>\n\\end{array}}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{g:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{1}{{x + 1}}}<br \/>\n\\end{array}}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{h:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to {x^2} &#8211; x}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<p><span style=\"color: #0000ff;\">\\[{f + g}\\]<\/span><\/p>\n<p>\\[{D_{f + g}} = {D_f} \\cap {D_g} = \\mathbb{R} \\cap \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} = \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\}\\]<\/p>\n<p>\\[\\left( {f + g} \\right)\\left( x \\right) = f\\left( x \\right) + g\\left( x \\right) = {x^2} + \\frac{1}{{x + 1}} = \\frac{{{x^3} + {x^2} + 1}}{{x + 1}}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{f + g:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{{x^3} + {x^2} + 1}}{{x + 1}}}<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<p><span style=\"color: #0000ff;\">\\[{f \\times g}\\]<\/span><\/p>\n<p>\\[{D_{f \\times g}} = {D_f} \\cap {D_g} = \\mathbb{R} \\cap \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} = \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\}\\]<\/p>\n<p>\\[\\left( {f \\times g} \\right)\\left( x \\right) = f\\left( x \\right) \\times g\\left( x \\right) = {x^2} \\times \\frac{1}{{x + 1}} = \\frac{{{x^2}}}{{x + 1}}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{f \\times g:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{{x^2}}}{{x + 1}}}<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<p><span style=\"color: #0000ff;\">\\[{\\frac{f}{g}}\\]<\/span><\/p>\n<p>\\[{D_{\\frac{f}{g}}} = {D_f} \\cap {D_g} \\cap \\left\\{ {x \\in \\mathbb{R}:g\\left( x \\right) \\ne 0} \\right\\} = \\mathbb{R} \\cap \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\cap \\mathbb{R} = \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\}\\]<\/p>\n<p>\\[\\frac{f}{g}\\left( x \\right) = \\frac{{f\\left( x \\right)}}{{g\\left( x \\right)}} = \\frac{{{x^2}}}{{\\frac{1}{{x + 1}}}} = {x^3} + {x^2}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{\\frac{f}{g}:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to {x^3} + {x^2}}<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<p><span style=\"color: #0000ff;\">\\[{h &#8211; g}\\]<\/span><\/p>\n<p>\\[{D_{h &#8211; g}} = {D_h} \\cap {D_g} = \\mathbb{R} \\cap \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} = \\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\}\\]<\/p>\n<p>\\[\\left( {h &#8211; g} \\right)\\left( x \\right) = h\\left( x \\right) &#8211; g\\left( x \\right) = {x^2} &#8211; x &#8211; \\frac{1}{{x + 1}} = \\frac{{{x^3} + {x^2} &#8211; {x^2} &#8211; x &#8211; 1}}{{x + 1}} = \\frac{{{x^3} &#8211; x &#8211; 1}}{{x + 1}}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{h &#8211; g:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{{x^3} &#8211; x &#8211; 1}}{{x + 1}}}<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<p><span style=\"color: #0000ff;\">\\[\\frac{f}{h}\\]<\/span><\/p>\n<p>\\[{D_{\\frac{f}{h}}} = {D_f} \\cap {D_h} \\cap \\left\\{ {x \\in \\mathbb{R}:h\\left( x \\right) \\ne 0} \\right\\} = \\mathbb{R} \\cap \\mathbb{R} \\cap \\mathbb{R}\\backslash \\left\\{ {0,1} \\right\\} = \\mathbb{R}\\backslash \\left\\{ {0,1} \\right\\}\\]<\/p>\n<p>\\[\\frac{f}{h}\\left( x \\right) = \\frac{{f\\left( x \\right)}}{{g\\left( x \\right)}} = \\frac{{{x^2}}}{{{x^2} &#8211; x}} = \\frac{{{x^2}}}{{x\\left( {x &#8211; 1} \\right)}} = \\frac{x}{{x &#8211; 1}}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{\\frac{f}{h}:}&amp;{\\mathbb{R}\\backslash \\left\\{ {0,1} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{x}{{x &#8211; 1}}}<br \/>\n\\end{array}\\]<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11808' onClick='GTTabs_show(0,11808)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere as fun\u00e7\u00f5es definidas por: \\[\\begin{array}{*{20}{r}} {\\begin{array}{*{20}{l}} {f:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\ {}&amp;{x \\to {x^2}} \\end{array}}&amp;{}&amp;{\\begin{array}{*{20}{l}} {g:}&amp;{\\mathbb{R}\\backslash \\left\\{ { &#8211; 1} \\right\\} \\to \\mathbb{R}} \\\\ {}&amp;{x \\to \\frac{1}{{x + 1}}} \\end{array}}&amp;{}&amp;{\\begin{array}{*{20}{l}} {h:}&amp;{\\mathbb{R}&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14114,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[98,97,147],"tags":[422,363,366,364,365],"series":[],"class_list":["post-11808","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-operacoes-com-funcoes","tag-11-o-ano","tag-funcao-diferenca","tag-funcao-produto","tag-funcao-quociente","tag-funcao-soma"],"views":1967,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat56.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11808","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=11808"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11808\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11808"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11808"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11808"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11808"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}