{"id":27097,"date":"2021-03-23T13:00:27","date_gmt":"2021-03-23T13:00:27","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27097"},"modified":"2024-06-06T22:07:33","modified_gmt":"2024-06-06T22:07:33","slug":"funcoes-algebricas-de-numeros-reais","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/funcoes-algebricas-de-numeros-reais\/","title":{"rendered":"Fun\u00e7\u00f5es Alg\u00e9bricas de N\u00fameros Reais"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px; text-align:center;\">\n<h1>Fun\u00e7\u00f5es Alg\u00e9bricas de N\u00fameros Reais<\/h1>\n<p>    <em><strong>Resumo:<\/strong><br \/>\nNesta aula, exploraremos as fun\u00e7\u00f5es alg\u00e9bricas, sua defini\u00e7\u00e3o, propriedades e aplica\u00e7\u00f5es. Essas fun\u00e7\u00f5es s\u00e3o fundamentais em diversas \u00e1reas da matem\u00e1tica e t\u00eam amplas aplica\u00e7\u00f5es pr\u00e1ticas.<\/em><\/p>\n<p>    <strong>OBJETIVOS DE APRENDIZAGEM<\/strong><\/p>\n<p>Ao finalizar esta aula, o estudante ser\u00e1 capaz de:<\/p>\n<p style=\"text-align:left;\">\n        1. Definir e compreender as fun\u00e7\u00f5es alg\u00e9bricas e suas propriedades.<br \/>\n        2. Identificar o dom\u00ednio e a imagem das fun\u00e7\u00f5es alg\u00e9bricas.<br \/>\n        3. Aplicar fun\u00e7\u00f5es alg\u00e9bricas em contextos matem\u00e1ticos e pr\u00e1ticos.\n    <\/p>\n<p>    <strong>\u00cdNDICE DE CONTE\u00daDOS:<\/strong><\/p>\n<p>\n        <a href=\"#1\"><strong>1. Introdu\u00e7\u00e3o<\/strong><\/a><br \/>\n        <a href=\"#2\"><strong>2. O que s\u00e3o Fun\u00e7\u00f5es Alg\u00e9bricas?<\/strong><\/a><br \/>\n        <a href=\"#3\"><strong>3. Outros Tipos de Fun\u00e7\u00f5es<\/strong><\/a>\n    <\/p>\n<p>    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/dLHAQhMjuag\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2><strong>1. Introdu\u00e7\u00e3o<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=dLHAQhMjuag&amp;t=173s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">O estudo das fun\u00e7\u00f5es alg\u00e9bricas come\u00e7a introduzindo as vari\u00e1veis:<\/span><\/strong><\/a> s\u00edmbolos que representam onde um n\u00famero pode estar. Tradicionalmente, as letras <span class=\"katex-eq\" data-katex-display=\"false\">x, y, z<\/span> s\u00e3o usadas para representar n\u00fameros reais, enquanto em outros contextos, <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span> \u00e9 preferida para n\u00fameros complexos. Tamb\u00e9m \u00e9 costume usar subscritos quando h\u00e1 muitas vari\u00e1veis. Assim, <span class=\"katex-eq\" data-katex-display=\"false\">x_1, x_2, \\cdots , x_n<\/span> tamb\u00e9m s\u00e3o exemplos de vari\u00e1veis.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    As fun\u00e7\u00f5es alg\u00e9bricas s\u00e3o fundamentais em diversas \u00e1reas da matem\u00e1tica e suas aplica\u00e7\u00f5es. Essas fun\u00e7\u00f5es s\u00e3o definidas por express\u00f5es alg\u00e9bricas que envolvem opera\u00e7\u00f5es b\u00e1sicas como adi\u00e7\u00e3o, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o, divis\u00e3o, pot\u00eancias e ra\u00edzes de vari\u00e1veis. Compreender as fun\u00e7\u00f5es alg\u00e9bricas \u00e9 essencial para o estudo de muitos ramos da matem\u00e1tica pura e aplicada, incluindo \u00e1lgebra, c\u00e1lculo, geometria e teoria dos n\u00fameros. Al\u00e9m disso, t\u00eam import\u00e2ncia crucial na f\u00edsica, engenharia, economia e ci\u00eancias sociais, pois permitem modelar e analisar fen\u00f4menos reais de maneira precisa e eficiente.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    No campo educacional, as fun\u00e7\u00f5es alg\u00e9bricas servem como uma base s\u00f3lida para o desenvolvimento do pensamento abstrato e habilidades de resolu\u00e7\u00e3o de problemas. Atrav\u00e9s do estudo dessas fun\u00e7\u00f5es, os estudantes aprendem a manipular express\u00f5es alg\u00e9bricas e a entender as rela\u00e7\u00f5es entre vari\u00e1veis, o que \u00e9 fundamental para avan\u00e7ar em matem\u00e1tica mais complexa.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Na vida cotidiana, as fun\u00e7\u00f5es alg\u00e9bricas s\u00e3o usadas em uma variedade de contextos pr\u00e1ticos. Por exemplo, s\u00e3o aplicadas na gest\u00e3o financeira para calcular juros e amortiza\u00e7\u00f5es, na inform\u00e1tica para desenvolver algoritmos e na engenharia para projetar estruturas e sistemas. As fun\u00e7\u00f5es alg\u00e9bricas tamb\u00e9m s\u00e3o essenciais na an\u00e1lise de dados e modelagem estat\u00edstica, ajudando a interpretar e prever comportamentos com base em dados observados.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Em resumo, o estudo das fun\u00e7\u00f5es alg\u00e9bricas n\u00e3o \u00e9 apenas uma pedra angular da matem\u00e1tica, mas tamb\u00e9m tem uma ampla gama de aplica\u00e7\u00f5es pr\u00e1ticas que destacam sua relev\u00e2ncia e utilidade no mundo moderno. Com uma compreens\u00e3o s\u00f3lida dessas fun\u00e7\u00f5es, problemas complexos podem ser abordados e solu\u00e7\u00f5es inovadoras podem ser desenvolvidas em diversos campos.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><strong>2. O que s\u00e3o Fun\u00e7\u00f5es Alg\u00e9bricas?<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=dLHAQhMjuag&amp;t=242s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">As fun\u00e7\u00f5es alg\u00e9bricas s\u00e3o um tipo especial de fun\u00e7\u00e3o.<\/span><\/strong><\/a> Uma fun\u00e7\u00e3o \u00e9 uma lei de correspond\u00eancia entre dois conjuntos que representamos atrav\u00e9s da nota\u00e7\u00e3o:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">f: A\\longmapsto B<\/span>\n<p style=\"text-align: justify; color: #000000;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u00e9 o conjunto de entrada e <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> \u00e9 o conjunto de sa\u00edda.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Toda fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> tamb\u00e9m tem um <strong>dom\u00ednio<\/strong> (<span class=\"katex-eq\" data-katex-display=\"false\">Dom(f)<\/span>) e uma <strong>imagem<\/strong> (<span class=\"katex-eq\" data-katex-display=\"false\">Rec(f)<\/span>). O dom\u00ednio \u00e9 o conjunto de todos os valores de entrada para os quais a fun\u00e7\u00e3o produz um resultado v\u00e1lido, e a imagem \u00e9 o conjunto de todas as sa\u00eddas poss\u00edveis da fun\u00e7\u00e3o. A imagem tamb\u00e9m \u00e9 chamada de <strong>Imagem<\/strong>, e o dom\u00ednio \u00e9 chamado de <strong>pr\u00e9-imagem.<\/strong> Ao definir uma fun\u00e7\u00e3o, \u00e0s vezes \u00e9 costume escrev\u00ea-la de qualquer uma das duas formas a seguir:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">f: Dom(f)\\subseteq A\\longmapsto Rec(f)\\subseteq B<\/span>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">f: Dom(f)\\longmapsto Rec(f)<\/span>\n<p style=\"text-align: justify; color: #000000;\">\n    Assim, as fun\u00e7\u00f5es alg\u00e9bricas s\u00e3o aquelas escritas em termos das <strong>opera\u00e7\u00f5es alg\u00e9bricas<\/strong> de suas vari\u00e1veis, como adi\u00e7\u00e3o, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o, divis\u00e3o, pot\u00eancia e raiz principal. Al\u00e9m disso, diz-se que uma fun\u00e7\u00e3o \u00e9 de vari\u00e1vel real se suas vari\u00e1veis forem substitu\u00eddas por n\u00fameros reais, de vari\u00e1vel complexa se forem substitu\u00eddas por n\u00fameros complexos, e assim por diante com qualquer outro conjunto num\u00e9rico. Tamb\u00e9m se fala de fun\u00e7\u00f5es de uma, duas, tr\u00eas ou m\u00faltiplas vari\u00e1veis, dependendo de terem uma, duas, tr\u00eas ou muitas vari\u00e1veis.\n<\/p>\n<h3>2.1. Exemplos de Fun\u00e7\u00f5es Alg\u00e9bricas<\/h3>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\n        <a href=\"https:\/\/www.youtube.com\/watch?v=dLHAQhMjuag&amp;t=349s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consideremos a seguinte fun\u00e7\u00e3o<\/span><\/strong><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\">\n            <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{matrix}\n\n            f : &amp; \\mathbb{R} &amp; \\longrightarrow &amp; \\mathbb{R} \\\\\n\n            &amp; x &amp; \\longmapsto &amp; f(x) =x^3 + 5x + \\displaystyle \\frac{6}{\\sqrt{x}} \\\\\n\n            \\end{matrix}\n\n            <\/span>\n        <\/p>\n<p style=\"text-align: justify; color: #000000;\">\n            Esta \u00e9 uma fun\u00e7\u00e3o alg\u00e9brica de uma vari\u00e1vel real. Aqui podemos ver diretamente que\n        <\/p>\n<p style=\"text-align: center; color: #000000;\">\n            <span class=\"katex-eq\" data-katex-display=\"false\">Dom(f) = \\{x\\in\\mathbb{R}\\;|\\; x\\gt 0\\} = ]0, +\\infty[ <\/span>\n        <\/p>\n<p style=\"text-align: justify; color: #000000;\">\n            Isso ocorre porque n\u00e3o h\u00e1 divis\u00f5es por zero e porque a raiz principal s\u00f3 est\u00e1 bem definida para n\u00fameros reais positivos.\n        <\/p>\n<\/li>\n<li>\n        <a href=\"https:\/\/www.youtube.com\/watch?v=dLHAQhMjuag&amp;t=707s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Agora vamos revisar a seguinte fun\u00e7\u00e3o<\/span><\/strong><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\">\n            <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{matrix}\n\n            f : &amp; \\mathbb{R}^2 &amp; \\longrightarrow &amp; \\mathbb{R} \\\\\n\n            &amp; (x,y) &amp; \\longmapsto &amp; f(x,y) =\\displaystyle \\frac{2xy + \\frac{3}{x^2}}{\\sqrt[3]{y-1}} \\\\\n\n            \\end{matrix}\n\n            <\/span>\n        <\/p>\n<p style=\"text-align: justify; color: #000000;\">\n            Esta \u00e9 uma fun\u00e7\u00e3o de duas vari\u00e1veis reais que resulta em um n\u00famero real. Isso tamb\u00e9m \u00e9 conhecido como <strong>campo escalar.<\/strong> Esse tipo de fun\u00e7\u00e3o est\u00e1 al\u00e9m do escopo deste curso, mas \u00e9 muito \u00fatil na f\u00edsica para descrever quantidades como temperatura ou distribui\u00e7\u00f5es de densidade. O dom\u00ednio dessa fun\u00e7\u00e3o tamb\u00e9m pode ser visto \u00aba olho nu\u00bb.\n        <\/p>\n<p style=\"text-align: center; color: #000000;\">\n            <span class=\"katex-eq\" data-katex-display=\"false\">Dom(f) = \\{(x,y)\\in\\mathbb{R}^2\\;|\\; x \\neq 0 \\wedge y\\neq 1 \\}<\/span>\n        <\/p>\n<\/li>\n<\/ol>\n<h3>2.2. Coment\u00e1rios sobre o Gr\u00e1fico e a Imagem<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    Determinar a imagem geralmente \u00e9 complicado. Posteriormente veremos t\u00e9cnicas que nos permitir\u00e3o fazer isso facilmente, mesmo em casos em que pareceria imposs\u00edvel alcan\u00e7ar algebricamente. No entanto, mesmo com esses m\u00e9todos, haver\u00e1 problemas porque \u00e0s vezes s\u00e3o necess\u00e1rias t\u00e9cnicas que v\u00e3o al\u00e9m do escopo deste curso, como o c\u00e1lculo de pontos cr\u00edticos para identificar m\u00e1ximos e m\u00ednimos no <strong>c\u00e1lculo diferencial.<\/strong> No entanto, mesmo sem o c\u00e1lculo, h\u00e1 muito que pode ser feito, e essas coisas ser\u00e3o abordadas em seu devido tempo.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Se voc\u00ea ainda estiver interessado em conhecer a imagem e o gr\u00e1fico dessas fun\u00e7\u00f5es, sempre pode recorrer ao Wolfram Alpha. V\u00e1 para <a href=\"https:\/\/www.wolframalpha.com\/\" rel=\"noopener, nofollow noopener\" target=\"_blank\">https:\/\/www.wolframalpha.com\/<\/a> e tente copiar e colar isto:<\/p>\n<p>    <code>x^3 + 5x + \\dfrac{6}{\\sqrt{x}}<\/code><\/p>\n<p>para ter uma ideia de como seria o primeiro exemplo. Para o segundo, copie e cole isto:<\/p>\n<p>    <code>\\dfrac{2xy + \\dfrac{3}{x^2}}{\\sqrt[3]{y-1}}<\/code>\n<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><strong>3. Outros Tipos de Fun\u00e7\u00f5es<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=dLHAQhMjuag&amp;t=1041s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">As fun\u00e7\u00f5es que estudamos neste curso podem ser divididas em dois tipos:<\/span><\/strong><\/a> Fun\u00e7\u00f5es Alg\u00e9bricas e Fun\u00e7\u00f5es Transcendentais. As fun\u00e7\u00f5es alg\u00e9bricas, como vimos, s\u00e3o aquelas que s\u00e3o escritas em termos das opera\u00e7\u00f5es fundamentais, enquanto as fun\u00e7\u00f5es transcendentais n\u00e3o podem ser escritas dessa maneira ou requerem express\u00f5es compostas de opera\u00e7\u00f5es infinitas. As fun\u00e7\u00f5es alg\u00e9bricas podem ser ainda divididas em dois tipos: fun\u00e7\u00f5es polinomiais e n\u00e3o polinomiais. Uma fun\u00e7\u00e3o polinomial \u00e9 qualquer fun\u00e7\u00e3o que pode ser escrita como a soma ou diferen\u00e7a de pot\u00eancias. Algo assim:\n<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(x) = \\sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + \\cdots + a_n x^n<\/span>\n<p style=\"text-align: justify; color: #000000;\">\n    Qualquer fun\u00e7\u00e3o que n\u00e3o seja dessa forma \u00e9 n\u00e3o-polinomial. Entre as fun\u00e7\u00f5es n\u00e3o-polinomiais, destacam-se as fun\u00e7\u00f5es racionais, que s\u00e3o aquelas que podem ser escritas como o quociente entre duas fun\u00e7\u00f5es polinomiais.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Fun\u00e7\u00f5es Alg\u00e9bricas de N\u00fameros Reais Resumo: Nesta aula, exploraremos as fun\u00e7\u00f5es alg\u00e9bricas, sua defini\u00e7\u00e3o, propriedades e aplica\u00e7\u00f5es. Essas fun\u00e7\u00f5es s\u00e3o fundamentais em diversas \u00e1reas da matem\u00e1tica e t\u00eam amplas aplica\u00e7\u00f5es pr\u00e1ticas. OBJETIVOS DE APRENDIZAGEM Ao finalizar esta aula, o estudante ser\u00e1 capaz de: 1. Definir e compreender as fun\u00e7\u00f5es alg\u00e9bricas e suas propriedades. 2. Identificar [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27079,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-27097","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-e-geometria","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Fun\u00e7\u00f5es Alg\u00e9bricas de N\u00fameros Reais - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubra as fun\u00e7\u00f5es alg\u00e9bricas de n\u00fameros reais, sua defini\u00e7\u00e3o, propriedades e aplica\u00e7\u00f5es. 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