{"id":11631,"date":"2013-02-21T14:09:37","date_gmt":"2013-02-21T14:09:37","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11631"},"modified":"2022-01-20T17:22:35","modified_gmt":"2022-01-20T17:22:35","slug":"as-funcoes-afins-f-g-e-h","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11631","title":{"rendered":"As fun\u00e7\u00f5es afins $f$, $g$ e $h$"},"content":{"rendered":"<p><ul id='GTTabs_ul_11631' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11631' class='GTTabs_curr'><a  id=\"11631_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11631' ><a  id=\"11631_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_11631'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_11635\" style=\"width: 201px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-11635\" data-attachment-id=\"11635\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11635\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\" data-orig-size=\"273,319\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&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\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\" class=\" wp-image-11635 \" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\" alt=\"Gr\u00e1ficos\" width=\"191\" height=\"223\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg 273w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh-256x300.jpg 256w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh-128x150.jpg 128w\" sizes=\"auto, (max-width: 191px) 100vw, 191px\" \/><\/a><p id=\"caption-attachment-11635\" class=\"wp-caption-text\">Gr\u00e1ficos das fun\u00e7\u00f5es $f$, $g$ e $h$<\/p><\/div>\n<p>No referencial da figura encontam-se representadas as fun\u00e7\u00f5es afins $f$, $g$ e $h$, definidas por:<\/p>\n<ul>\n<li>$f\\left( x \\right) = 3x &#8211; 6$<\/li>\n<li>$g\\left( x \\right) =\u00a0 &#8211; 0,5x + 1,5$<\/li>\n<li>$h\\left( x \\right) = 1,5$<\/li>\n<\/ul>\n<ol>\n<li>Relacione os gr\u00e1ficos com as fun\u00e7\u00f5es dadas.<\/li>\n<li>Determine os zeros de $f$\u00a0e de $g$.<\/li>\n<li>Calcule a \u00e1rea dos tri\u00e2ngulos [CDB] e [ABP].<\/li>\n<li>Determine, sob a forma de intervalo, os conjuntos de valores de $x$ para os quais:\n<p>a) a fun\u00e7\u00e3o $f$ \u00e9 positiva;<\/p>\n<p>b) a fun\u00e7\u00e3o $g$ \u00e9 negativa;<\/p>\n<p>c) $f\\left( x \\right) \\geqslant g\\left( x \\right)$;<\/p>\n<p>d) $f\\left( x \\right) &lt; 1,5$.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11631' onClick='GTTabs_show(1,11631)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11631'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11635\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11635\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\" data-orig-size=\"273,319\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&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\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\" class=\"alignright  wp-image-11635\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg\" alt=\"Gr\u00e1ficos\" width=\"191\" height=\"223\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh.jpg 273w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh-256x300.jpg 256w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/fgh-128x150.jpg 128w\" sizes=\"auto, (max-width: 191px) 100vw, 191px\" \/><\/a><br \/>\n$r$ \u2192 $h\\left( x \\right) = 1,5$<\/p>\n<p>$s$ \u2192 $g\\left( x \\right) =\u00a0 &#8211; 0,5x + 1,5$<\/p>\n<p>$t$ \u2192 $f\\left( x \\right) = 3x &#8211; 6$<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Como $$f\\left( x \\right) = 0 \\Leftrightarrow 3x &#8211; 6 = 0 \\Leftrightarrow x = 2$$ e $$g\\left( x \\right) = 0 \\Leftrightarrow\u00a0 &#8211; 0,5x + 1,5 = 0 \\Leftrightarrow x = 3$$ ent\u00e3o $x = 2$ \u00e9 o \u00fanico zero de $f$ e $x = 3$ \u00e9 o \u00fanico zero de $g$.<br \/>\n\u00ad<\/li>\n<li>Os pontos $C$ e $D$ s\u00e3o pontos de ordenada nula, pois pertencem ao eixo $Ox$.<br \/>\nPor outro lado, como s\u00e3o pontos das retas $t$ e $s$, respetivamente, as suas abcissas s\u00e3o os zeros das fun\u00e7\u00f5es $f$ e $g$. Assim, $C\\left( {2,0} \\right)$ e $D\\left( {3,0} \\right)$.<\/p>\n<p>O ponto $B$ \u00e9 a interse\u00e7\u00e3o das retas $s$ e $t$:<br \/>\n$$\\begin{array}{*{20}{c}}<br \/>\n{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{y = 3x &#8211; 6} \\\\<br \/>\n{y =\u00a0 &#8211; 0,5x + 1,5}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{3x &#8211; 6 =\u00a0 &#8211; 0,5x + 1,5} \\\\<br \/>\n{y = 3x &#8211; 6}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{3,5x = 7,5} \\\\<br \/>\n{y =\u00a0 &#8211; 0,5x + 1,5}<br \/>\n\\end{array}} \\right.}&amp; \\Leftrightarrow &amp;{\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x = \\frac{{15}}{7}} \\\\<br \/>\n{y = \\frac{3}{7}}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}$$<br \/>\nPortanto, $B\\left( {\\frac{{15}}{7},\\frac{3}{7}} \\right)$.<\/p>\n<p>Assim, sendo $B&#8217;$ a proje\u00e7\u00e3o ortogonal do ponto $B$ sobe o eixo Ox, temos: $${A_{\\left[ {CDB} \\right]}} = \\frac{{\\overline {CD}\u00a0 \\times \\overline {BB&#8217;} }}{2} = \\frac{{1 \\times \\frac{3}{7}}}{2} = \\frac{3}{{14}}$$<\/p>\n<\/li>\n<li>Os pontos $A$ e $P$ t\u00eam ordenada $1,5$, pois s\u00e3o pontos de reta $r$.\n<p>Como o ponto $A$ pertence tamb\u00e9m \u00e0 reta $s$, a sua abcissa \u00e9 $1,5 =\u00a0 &#8211; 0,5x + 1,5 \\Leftrightarrow x = 0$. Logo, $A\\left( {0;1,5} \\right)$.<\/p>\n<p>Como o ponto $P$ pertence tamb\u00e9m \u00e0 reta $t$, a sua abcissa \u00e9 $1,5 = 3x &#8211; 6 \\Leftrightarrow x = 2,5$. Logo, $P\\left( {2,5;1,5} \\right)$.<\/p>\n<p>Assim, sendo $B&#8221;$ a proje\u00e7\u00e3o ortogonal do ponto $B$ sobe a reta $r$, temos: $${A_{\\left[ {ABP} \\right]}} = \\frac{{\\overline {AP}\u00a0 \\times \\overline {BB&#8221;} }}{2} = \\frac{{\\frac{5}{2} \\times \\left( {\\frac{3}{2} &#8211; \\frac{3}{7}} \\right)}}{2} = \\frac{{\\frac{5}{2} \\times \\left( {\\frac{{21}}{{14}} &#8211; \\frac{6}{{14}}} \\right)}}{2} = \\frac{{\\frac{5}{2} \\times \\frac{{15}}{{14}}}}{2} = \\frac{{75}}{{56}}$$<\/p>\n<\/li>\n<li>a) $$\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) &gt; 0}&amp; \\Leftrightarrow &amp;{3x &#8211; 6 &gt; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x &gt; 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left] {2, + \\infty } \\right[}<br \/>\n\\end{array}$$<br \/>\nb) $$\\begin{array}{*{20}{l}}<br \/>\n{g\\left( x \\right) &lt; 0}&amp; \\Leftrightarrow &amp;{ &#8211; 0,5x + 1,5 &lt; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{ &#8211; 0,5x &lt;\u00a0 &#8211; 1,5} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x &gt; 3} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left] {3, + \\infty } \\right[}<br \/>\n\\end{array}$$<br \/>\nc) $$\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) \\geqslant g\\left( x \\right)}&amp; \\Leftrightarrow &amp;{3x &#8211; 6 \\geqslant\u00a0 &#8211; 0,5x + 1,5} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{3,5x \\geqslant 7,5} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\geqslant \\frac{{15}}{7}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left[ {\\frac{{15}}{7}, + \\infty } \\right[}<br \/>\n\\end{array}$$<br \/>\nd) $$\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) &lt; 1,5}&amp; \\Leftrightarrow &amp;{3x &#8211; 6 &lt; 1,5} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x &lt; 2,5} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\left] { &#8211; \\infty ;2,5} \\right[}<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11631' onClick='GTTabs_show(0,11631)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado No referencial da figura encontam-se representadas as fun\u00e7\u00f5es afins $f$, $g$ e $h$, definidas por: $f\\left( x \\right) = 3x &#8211; 6$ $g\\left( x \\right) =\u00a0 &#8211; 0,5x + 1,5$ $h\\left(&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20749,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[321,97,343],"tags":[429,345,344],"series":[],"class_list":["post-11631","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10-o-ano","category-aplicando","category-funcoes-e-graficos","tag-10-o-ano","tag-funcao-afim","tag-grafico"],"views":3315,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2013\/02\/10V2Pag037-6_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11631","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=11631"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11631\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20749"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11631"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11631"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11631"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}