{"id":131999,"date":"2024-12-11T09:50:59","date_gmt":"2024-12-11T12:50:59","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=131999"},"modified":"2025-02-19T15:08:51","modified_gmt":"2025-02-19T18:08:51","slug":"equacao-funcional-definicao-tipos-e-aplicacoes","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/","title":{"rendered":"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Na <span id=\"urn:enhancement-de060eb3-f941-43fd-80d3-b740b9f18846\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>, temos diversos tipos de conte&uacute;do, a <span id=\"urn:local-annotation-411441\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao-funcional-de-cauchy\">equa&ccedil;&atilde;o funcional<\/span> &eacute; um exemplo deles que envolve <span id=\"urn:enhancement-36bfef32-b800-4b57-a25c-ee6c3f529114\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/conhecimento\">conhecimento<\/span> de v&aacute;rios conceitos, como fun&ccedil;&otilde;es, substitui&ccedil;&otilde;es, teoria de <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/conjuntos\/\">conjuntos<\/a>, aplica&ccedil;&atilde;o de diversas propriedades, entre outros.<p class=\"wp-block-paragraph\">Em vista disso, ela &eacute; usualmente vista nas olimp&iacute;adas de <span id=\"urn:enhancement-aba256d2-338f-42be-98b1-2352d46b5cf2\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>, j&aacute; que demanda a aplica&ccedil;&atilde;o de diversos tipos de <span id=\"urn:enhancement-6569ffdc-6aeb-4908-897c-75d33a15debb\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/conhecimento\">conhecimentos<\/span> para resolver os problemas. Veja no artigo do <span id=\"urn:enhancement-b4a56153-5939-44d3-b081-025fe8a6f52a\" class=\"textannotation disambiguated wl-organization\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/estrategia_vestibulares\">Estrat&eacute;gia Vestibulares<\/span> mais detalhes sobre o que &eacute; <span id=\"urn:enhancement-298874d6-9609-4a2e-89b2-1c60c8d3c889\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> funcional, como identificar e algumas de suas aplica&ccedil;&otilde;es.&nbsp;&nbsp;<\/p><p class=\"wp-block-paragraph\"><div class=\"cta-medio\" style=\"background-image: url(https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/08\/curso-enem-1-estrategia-vestibulares-ctamedio.jpg);\"><h3>Pacote Enem TOP <\/h3> <p>Estude com o Estrat\u00e9gia Vestibulares<\/p><div class=\"cta-botao\"><a id=\"cta-medio\" href=\"https:\/\/vestibulares.estrategia.com\/curso\/enem-top\/\" target=\"_blank\" style=\"background:rgb(255,150,0)\">Saiba mais<\/a><\/div><\/div>\n\n\n\n<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Conceitos-basicos-das-equacoes-funcionais\" >Conceitos b&aacute;sicos das equa&ccedil;&otilde;es funcionais<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Dominio-e-Contradominio\" >Dom&iacute;nio e Contradom&iacute;nio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Tipos-de-equacoes-funcionais\" >Tipos de equa&ccedil;&otilde;es funcionais<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Estrategias-de-resolucao-de-equacoes-funcionais\" >Estrat&eacute;gias de resolu&ccedil;&atilde;o de equa&ccedil;&otilde;es funcionais<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Inducao-matematica\" >Indu&ccedil;&atilde;o matem&aacute;tica:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Exemplos-e-aplicacoes\" >Exemplos e aplica&ccedil;&otilde;es:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#Estuda-com-a-Coruja-para-suas-provas\" >Estuda com a Coruja para suas provas!<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-conceitos-basicos-das-equacoes-funcionais\"><span class=\"ez-toc-section\" id=\"Conceitos-basicos-das-equacoes-funcionais\"><\/span>Conceitos b&aacute;sicos das equa&ccedil;&otilde;es funcionais<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Quando vamos resolver uma simples <span id=\"urn:enhancement-0206afba-c3ac-4615-9551-ab75eb49d064\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span>, chegamos a um resultado que depende de uma vari&aacute;vel <em>x<\/em>, por exemplo. As <strong><span id=\"urn:enhancement-02a64a6c-9c6a-4a51-93b6-063a274e7dbf\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais<\/strong> s&atilde;o aquelas que, no sistema de <span id=\"urn:enhancement-62bd3874-9496-4b80-8247-b519507a1f55\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> formado quando estamos resolvendo o problema, as vari&aacute;veis desse <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/sistemas-lineares\/\">sistema<\/a> s&atilde;o fun&ccedil;&otilde;es, elas podem ser fun&ccedil;&otilde;es conhecidas ou fun&ccedil;&otilde;es desconhecidas, e pretendemos encontrar as fun&ccedil;&otilde;es desconhecidas como resposta.<\/p><p class=\"wp-block-paragraph\">Para resolver este tipo de problema, &eacute; necess&aacute;rio <strong>encontrar fun&ccedil;&otilde;es que satisfa&ccedil;am a <span id=\"urn:enhancement-bc4bead5-ea6f-4996-9d43-1c6fdb36bf1d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span><\/strong>. &Eacute; importante testar todas as solu&ccedil;&otilde;es encontradas, pois pode aparecer uma fun&ccedil;&atilde;o que n&atilde;o satisfa&ccedil;a a <span id=\"urn:enhancement-d07ff1e3-94a1-4684-96ce-93383109e2db\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> no momento da resolu&ccedil;&atilde;o.<\/p><p class=\"wp-block-paragraph\">Ao solucionar uma <span id=\"urn:enhancement-48280bb8-490e-4e7b-ae54-38933aa8df6e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> funcional, &eacute; necess&aacute;rio <span id=\"urn:enhancement-3c727da2-4a6a-4e30-97cf-e399aa27402b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/conhecimento\">conhecimentos<\/span> b&aacute;sicos relacionados tanto a fun&ccedil;&otilde;es, quanto a <span id=\"urn:enhancement-8d97d8c3-2f72-4a0f-a1b1-a00aa0622049\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span>. Diferenciamos as <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacoes\/\">equa&ccedil;&otilde;es alg&eacute;bricas<\/a> comuns das <span id=\"urn:enhancement-c513958e-8a29-4d17-9f88-cee1195fd7a3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais da seguinte <span id=\"urn:enhancement-56a81e31-019d-423e-8229-562532848b24\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span>:<\/p><ul class=\"wp-block-list\">\n<li>Equa&ccedil;&atilde;o alg&eacute;brica &eacute; definida com um conjunto de opera&ccedil;&otilde;es com uma cole&ccedil;&atilde;o de termos que envolvem uma inc&oacute;gnita <em>x<\/em>, por exemplo: <em><strong>x<sup>2<\/sup> + 7x + 16 = x &ndash; (7 + x)<sup>2<\/sup><\/strong> <\/em>ou<em> <strong>2x + 3 = 0<\/strong><\/em>;<\/li>\n\n\n\n<li>Equa&ccedil;&otilde;es funcionais s&atilde;o compostas por vari&aacute;veis e fun&ccedil;&otilde;es, assim sendo formada como uma express&atilde;o que resulta em uma fun&ccedil;&atilde;o, por exemplo: <em><strong>f(x-1) + 2 = x * f(x) <\/strong><\/em>ou<em><strong> f(f(x)+y) = 2f(x2)+3f(xy)<\/strong><\/em>.&nbsp;<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-dominio-e-contradominio\"><span class=\"ez-toc-section\" id=\"Dominio-e-Contradominio\"><\/span>Dom&iacute;nio e Contradom&iacute;nio<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\"><span id=\"urn:local-annotation-846955\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/dominio\">Dom&iacute;nio<\/span> e <span id=\"urn:enhancement-75003307-698c-4f6c-9b73-0abbd2611a6d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/contradominio\">contradom&iacute;nio<\/span> s&atilde;o conceitos constru&iacute;dos a partir da defini&ccedil;&atilde;o de fun&ccedil;&atilde;o, e s&atilde;o utilizados na resolu&ccedil;&atilde;o de <span id=\"urn:enhancement-6d155e33-9def-48e1-9dd9-91a7118ea92e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais para determinar o conjunto de valores que as fun&ccedil;&otilde;es e as vari&aacute;veis podem assumir.<\/p><p class=\"wp-block-paragraph\">Definindo uma fun&ccedil;&atilde;o, temos que, dado dois conjuntos <em>A<\/em> e <em>B<\/em>, a fun&ccedil;&atilde;o <em>f: A&rarr;B<\/em> associa cada elemento do conjunto <em>A<\/em> com algum elemento do conjunto <em>B<\/em>.&nbsp;<\/p><p class=\"wp-block-paragraph\">Por exemplo, se <em>A = {1,2,3}<\/em> e <em>B = {a,b,c,d}<\/em>, definindo a fun&ccedil;&atilde;o<em> f: A&rarr;B<\/em> como<em> f(1)=a<\/em>, <em>f(2)=c<\/em>, <em>f(3)=b<\/em>. Temos que o conjunto <em>A<\/em> &eacute; <strong>dom&iacute;nio<\/strong> da fun&ccedil;&atilde;o <em>f<\/em> e todo o conjunto <em>B<\/em> ser&aacute; o <strong><span id=\"urn:enhancement-a71eda8b-485e-46bd-a609-736cf8b9656a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/contradominio\">contradom&iacute;nio<\/span><\/strong> da fun&ccedil;&atilde;o <em>f<\/em>. J&aacute; a imagem da fun&ccedil;&atilde;o ser&aacute; os elementos do conjunto <em>B<\/em> que est&atilde;o associados aos elementos do conjunto <em>A<\/em>.<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh7-rt.googleusercontent.com\/docsz\/AD_4nXeBTZ3oQ-k_xi2--Q3-EF227lQ9vkoejKF9euLHSWObYmw-BXUOmQAp9L7tDOcJbCg6ikrdhDrGmaE49Or3rF_8G2TeNRe-mW6HAIrez56410OSCzq4YqBhmbKpDz3f3Vuo_Kfucy0uIikAad7zTdJox6Rs?key=hZAwkmZzN2B-R8kzmooAKA\" alt=\"\"><\/figure><p class=\"wp-block-paragraph\">Ainda sobre fun&ccedil;&otilde;es, a propriedade de <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-injetora-e-sobrejetora\/\">sobrejetividade<\/a><strong> <\/strong>da fun&ccedil;&atilde;o diz respeito a quando o conjunto <span id=\"urn:enhancement-220d363d-8b3e-43d7-b739-57247555efc5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/contradominio\">contradom&iacute;nio<\/span> &eacute; igual ao conjunto Imagem, e a propriedade de <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-injetora-e-sobrejetora\/\">injetividade<\/a><strong> <\/strong>&eacute; verdadeira quando cada elemento do dom&iacute;nio possui uma imagem diferente e, por fim, a propriedade da <a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/funcao-bijetora\/\">bijetividade <\/a>ocorre quando a fun&ccedil;&atilde;o &eacute; injetora e sobrejetora ao mesmo tempo.<\/p><h2 class=\"wp-block-heading\" id=\"h-tipos-de-equacoes-funcionais\"><span class=\"ez-toc-section\" id=\"Tipos-de-equacoes-funcionais\"><\/span>Tipos de equa&ccedil;&otilde;es funcionais<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Ao tratarmos sobre <span id=\"urn:enhancement-c3b950c3-ef5a-4ae8-906e-322985187be9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais, podemos nos deparar com diversos tipos. As principais categorias s&atilde;o aditivas, multiplicativas, Cauchy e Jensen.&nbsp;<\/p><h3 class=\"wp-block-heading\" id=\"h-equacoes-aditivas\">Equa&ccedil;&otilde;es aditivas<\/h3><p class=\"wp-block-paragraph\">As <span id=\"urn:enhancement-75257a9d-abf3-4535-8f6a-d3b31ec1f4d9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> aditivas s&atilde;o fun&ccedil;&otilde;es do tipo<strong> <em>f(x+y) = f(x) + f(y)<\/em><\/strong>&nbsp; para todo <em>x<\/em> e <em>y<\/em> pertencente ao conjunto dos Reais, isso significa que, dado dois valores <em>x<\/em> e <em>y<\/em>, a fun&ccedil;&atilde;o da <span id=\"urn:enhancement-73d3f09f-648a-4056-9b89-e9dcd6e52eb8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> desses dois valores &eacute; a igual a fun&ccedil;&atilde;o do primeiro valor somada a fun&ccedil;&atilde;o do segundo valor.&nbsp;<\/p><p class=\"wp-block-paragraph\">Observa-se tamb&eacute;m que a partir da aditividade, &eacute; poss&iacute;vel notar que essa <span id=\"urn:enhancement-af1c4c9e-9332-44d9-a85d-893be28a2ed2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> pode ser repetida mais de uma vez, d&aacute; seguinte <span id=\"urn:enhancement-dff2af71-d92c-49ba-9b96-11ccce5f8eaf\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span>:<\/p><p class=\"wp-block-paragraph\"><em><strong>f(x<sub>1<\/sub>+x<sub>2<\/sub>+&hellip;+x<sub>k<\/sub>) = f(x<sub>1<\/sub>)+f(x<sub>2<\/sub>)+&hellip;+f(x<sub>k<\/sub>) = k*f(x) = f(kx)<\/strong><\/em><\/p><p class=\"wp-block-paragraph\">onde o <em>x<\/em> &eacute; um valor, e o mesmo <em>x<\/em> &eacute; somado uma quantidade<em> k<\/em> de vezes. Um exemplo pr&aacute;tico: <\/p><p class=\"wp-block-paragraph\"><em>Dado f(x) + f(y) = y*f(x)<\/em>, vamos determinar as fun&ccedil;&otilde;es reais que obedecem essa <span id=\"urn:enhancement-bcd97a3a-ba25-4aa9-91b8-8b40597793b8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">regra<\/span>. <\/p><p class=\"wp-block-paragraph\">Assumindo <em>x = n &ndash; f(0) <\/em>e <em>y = 0<\/em>, temos que:<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"847\" height=\"39\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-4.png\" alt=\"\" class=\"wp-image-132038\" style=\"width:494px;height:auto\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-4.png 847w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-4-768x35.png 768w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-4-800x37.png 800w\" sizes=\"(max-width: 847px) 100vw, 847px\" \/><\/figure><p class=\"wp-block-paragraph\">Logo,<em> f(x) = 0<\/em> para qualquer valor de <em>x<\/em> &eacute; a solu&ccedil;&atilde;o<\/p><h3 class=\"wp-block-heading\" id=\"h-equacoes-multiplicativas\">Equa&ccedil;&otilde;es multiplicativas<\/h3><p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-7a097ac9-063c-471a-b294-0040985f5b5f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> multiplicativa &eacute; da <span id=\"urn:enhancement-c7fdf6e4-6547-4fb5-aa31-95a2e88e71e2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span><em> <strong>f(xy) = f(x) * f(y)<\/strong><\/em> para<em> x<\/em> e <em>y <\/em>pertencendo ao conjunto dos Reais, ou seja, a fun&ccedil;&atilde;o do produto de dois valores, &eacute; igual a fun&ccedil;&atilde;o de um valor multiplicado pela fun&ccedil;&atilde;o de outro valor. Disso podemos tirar que, quando <em>f(k) = k<\/em> temos que: <\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"641\" height=\"36\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-5.png\" alt=\"\" class=\"wp-image-132039\" style=\"width:404px;height:auto\"><\/figure><h3 class=\"wp-block-heading\" id=\"h-equacao-funcional-de-cauchy\">Equa&ccedil;&atilde;o funcional de Cauchy<\/h3><p class=\"wp-block-paragraph\">A <span id=\"urn:enhancement-8d5e6c6a-85fe-468b-bda1-c8630d343b36\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> aditiva de Cauchy &eacute; a <span id=\"urn:enhancement-ff3ab384-adf0-4a23-af7b-a14b7b64c1f0\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> do tipo <em><strong>f(x+y) = f(x) + f(y)<\/strong><\/em>&nbsp; para qualquer <em>x<\/em> e <em>y <\/em>pertencentes ao conjunto<em> &#8474;<\/em> (racionais), semelhante a aditiva, sua resolu&ccedil;&atilde;o gera informa&ccedil;&otilde;es muito importantes para conseguir fazer determinadas substitui&ccedil;&otilde;es nas <span id=\"urn:enhancement-2569ecae-fd8d-432a-9d48-154aed78a37e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&otilde;es<\/span>.<\/p><p class=\"wp-block-paragraph\">Para <em>f(x) = ax<\/em>, para qualquer <em>x<\/em> e qualquer <em>a<\/em> do conjunto dos n&uacute;meros racionais, temos como solu&ccedil;&atilde;o para <em>f(1) = a<\/em>.&nbsp;<\/p><p class=\"wp-block-paragraph\">Note que, se fizermos <em>f(x + 1) = f(x) + f(1) = f(x) + a<\/em>, da mesma <span id=\"urn:enhancement-14801de5-c911-43d3-97c2-5220413dd3aa\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> que <em>f(-(x + 1)) = f(-x) + f(-1)<\/em>.<\/p><p class=\"wp-block-paragraph\">Pela <span id=\"urn:enhancement-cc428750-808a-4367-b334-4a686115440d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/axiomas\">afirma&ccedil;&atilde;o<\/span> acima se fizermos <em>f(0)<\/em> temos como resultado:<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"596\" height=\"37\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-6.png\" alt=\"\" class=\"wp-image-132040\" style=\"width:405px;height:auto\"><\/figure><p class=\"wp-block-paragraph\">Dessa <span id=\"urn:enhancement-4ecf6d46-5cd0-4905-9844-15d1d28b19fe\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> provamos que <em>f(x) = ax<\/em><\/p><h3 class=\"wp-block-heading\" id=\"h-equacao-funcional-de-jensen\">Equa&ccedil;&atilde;o funcional de Jensen<\/h3><p class=\"wp-block-paragraph\">Jensen foi outro matem&aacute;tico que estudou <span id=\"urn:enhancement-9421dfb5-b6f5-486b-bd21-eb2f9b88eb92\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais e que n&atilde;o &eacute; muito conhecido, sua <span id=\"urn:enhancement-dfc084f8-3282-42d7-9b40-aa9ae144b760\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> &eacute; dada pela <span id=\"urn:enhancement-21e53701-fce2-4568-8a56-ec957bdd464d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/expressoes-algebricas\">express&atilde;o<\/span>:<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"378\" height=\"49\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-7.png\" alt=\"\" class=\"wp-image-132041\" style=\"width:255px;height:auto\"><\/figure><p class=\"wp-block-paragraph\">Ela sobre a fun&ccedil;&atilde;o aplicada na <span id=\"urn:enhancement-613bcc61-b7ae-43f3-bcad-afba38c1159c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/medias\">m&eacute;dia<\/span> de dois valores, ser igual a <span id=\"urn:enhancement-99cd3232-3158-4c90-9567-84d9b3ce1899\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/medias\">m&eacute;dia<\/span> de duas fun&ccedil;&otilde;es.<\/p><h2 class=\"wp-block-heading\" id=\"h-estrategias-de-resolucao-de-equacoes-funcionais\"><span class=\"ez-toc-section\" id=\"Estrategias-de-resolucao-de-equacoes-funcionais\"><\/span>Estrat&eacute;gias de resolu&ccedil;&atilde;o de equa&ccedil;&otilde;es funcionais<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Os problemas que envolvem <span id=\"urn:enhancement-42859499-f1ba-44ab-ad74-263743bb96ce\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais s&atilde;o muito utilizados em olimp&iacute;adas de <span id=\"urn:enhancement-30f47b04-7201-49cb-8f04-1b1f6db65dd4\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/matematica\">matem&aacute;tica<\/span>, j&aacute; que n&atilde;o possuem um m&eacute;todo definitivo de resolu&ccedil;&atilde;o e demandam bastante treino. Ent&atilde;o, aprender a manipular a <span id=\"urn:enhancement-7b2462f1-4b3a-48b7-bca7-365e7796e1b8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> a fim de obter informa&ccedil;&otilde;es sobre a fun&ccedil;&atilde;o &eacute; fundamental para resolver este tipo de problema. Algumas das estrat&eacute;gias de substitui&ccedil;&atilde;o inteligente mais adotadas s&atilde;o:<\/p><ul class=\"wp-block-list\">\n<li><strong>Observar o dom&iacute;nio e contradom&iacute;nio<\/strong> das fun&ccedil;&otilde;es envolvidas que ser&atilde;o determinantes para a resolu&ccedil;&atilde;o dos problemas;<\/li>\n\n\n\n<li>A <strong>substitui&ccedil;&atilde;o de valores<\/strong> <strong>espec&iacute;ficos<\/strong> para a vari&aacute;vel, por exemplo <em>-1<\/em>,<em> 0 <\/em>ou<em> 1<\/em>, para que, dependendo da equa&ccedil;&atilde;o, possam ser canceladas ou simplificadas;<\/li>\n\n\n\n<li>Fazer <strong>substitui&ccedil;&otilde;es de modo a se obter uma &uacute;nica vari&aacute;vel<\/strong> na equa&ccedil;&atilde;o, como por exemplo fazer <em>x=y<\/em> e <em>x=-z<\/em>;<\/li>\n\n\n\n<li><strong>Substitui&ccedil;&atilde;o para conseguir cancelamento de vari&aacute;veis<\/strong> na equa&ccedil;&atilde;o &eacute; fazer <em>y=-x<\/em>; e<\/li>\n\n\n\n<li><strong>Colocar uma express&atilde;o em fun&ccedil;&atilde;o de uma nova vari&aacute;vel<\/strong> caso seja necess&aacute;rio, como por exemplo se tivermos <em>5(f(f(x))<\/em>, far&iacute;amos a substitui&ccedil;&atilde;o <em>g = f(x)<\/em>, obtendo assim a express&atilde;o <em>5 * f(g)<\/em>.<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-inducao-matematica\"><span class=\"ez-toc-section\" id=\"Inducao-matematica\"><\/span>Indu&ccedil;&atilde;o matem&aacute;tica:<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">A indu&ccedil;&atilde;o matem&aacute;tica tamb&eacute;m est&aacute; presente para a resolu&ccedil;&atilde;o de <span id=\"urn:enhancement-edb7b75e-0915-4108-99ec-299e2385d4bb\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&otilde;es<\/span> funcionais, dominar essa t&eacute;cnica para provar que uma poss&iacute;vel solu&ccedil;&atilde;o &eacute; v&aacute;lida para todos os valores do dom&iacute;nio &eacute; essencial para fazer a prova real da solu&ccedil;&atilde;o encontrada.<\/p><p class=\"wp-block-paragraph\">Por indu&ccedil;&atilde;o, &eacute; poss&iacute;vel verificar que uma solu&ccedil;&atilde;o &eacute; v&aacute;lida <strong>provando para <em>f(n+1)<\/em> a partir de<em> f(n)<\/em> <\/strong>com <em>n<\/em> pertencente ao conjunto dos naturais, depois dos racionais, por fim para os reais provando que a fun&ccedil;&atilde;o &eacute; cont&iacute;nua no intervalo.&nbsp;<\/p><p class=\"wp-block-paragraph\">&Eacute; poss&iacute;vel construir a solu&ccedil;&atilde;o usando casos base e as propriedades de fun&ccedil;&otilde;es para encontrar resultados corretos, utilizamos essa t&eacute;cnica quando falamos sobre a <span id=\"urn:enhancement-f76afb99-64b3-428b-baea-a590ab7b5134\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> de Cauchy.&nbsp;<\/p><h2 class=\"wp-block-heading\" id=\"h-exemplos-e-aplicacoes\"><span class=\"ez-toc-section\" id=\"Exemplos-e-aplicacoes\"><\/span>Exemplos e aplica&ccedil;&otilde;es:<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">A seguir vamos ver alguns exemplos e utilizar a <span id=\"urn:enhancement-53997381-e5ea-47c5-b23e-344d7fa90d71\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/equacao\">equa&ccedil;&atilde;o<\/span> funcional de Cauchy, que &eacute; um exemplo cl&aacute;ssico para entender a como fazer substitui&ccedil;&otilde;es de <span id=\"urn:enhancement-9e8e7c10-9f8e-490b-953f-f9c5abacf877\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> de encontrar a solu&ccedil;&atilde;o geral.<\/p><p class=\"wp-block-paragraph\">Supondo a seguinte situa&ccedil;&atilde;o: uma fun&ccedil;&atilde;o aditiva <em>f: &#8477;&rarr;&#8477;<\/em> tal que <em>x = y =0<\/em>, temos que:<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"692\" height=\"33\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-8.png\" alt=\"\" class=\"wp-image-132042\" style=\"width:467px;height:auto\"><\/figure><p class=\"wp-block-paragraph\">Tamb&eacute;m podemos fazer a substitui&ccedil;&atilde;o<em> y=-x<\/em>, assim obtendo:<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"864\" height=\"34\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-9.png\" alt=\"\" class=\"wp-image-132043\" style=\"width:569px;height:auto\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-9.png 864w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-9-768x30.png 768w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-9-800x31.png 800w\" sizes=\"(max-width: 864px) 100vw, 864px\" \/><\/figure><p class=\"wp-block-paragraph\">essa &eacute; a demonstra&ccedil;&atilde;o que qualquer fun&ccedil;&atilde;o aditiva &eacute; &iacute;mpar.<\/p><p class=\"wp-block-paragraph\">Outro exemplo as fun&ccedil;&otilde;es para qualquer <em>x <\/em>e <em>y<\/em> em que <em>f(xy) = x*(fx) + y*(f(y)<\/em>, se <em>y = 1<\/em>, temos que:<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"842\" height=\"36\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-10.png\" alt=\"\" class=\"wp-image-132044\" style=\"width:545px;height:auto\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-10.png 842w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-10-768x33.png 768w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-10-800x34.png 800w\" sizes=\"(max-width: 842px) 100vw, 842px\" \/><\/figure><p class=\"wp-block-paragraph\">Se <em>x = 1<\/em>, <em>f(1) = 0<\/em>, da&iacute; temos que para quaisquer valores de <em>x<\/em>, <em>f(x) = 0<\/em>, pois <\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"354\" height=\"31\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-11.png\" alt=\"\" class=\"wp-image-132045\" style=\"width:252px;height:auto\"><\/figure><p class=\"wp-block-paragraph\">Aplicando o exemplo a uma quest&atilde;o, se tivermos <em>f: &#8477;&rarr;&#8477;<\/em> e <em>f(x + y) = x + f(y)<\/em> e que <em>f(0) = 2<\/em> e quisermos o valor de <em>f(100)<\/em>, resolvermos da seguinte maneira:<\/p><p class=\"wp-block-paragraph\">Como j&aacute; temos<em> f(0) = 2<\/em>, atribu&iacute;mos <em>x = 100<\/em> e <em>y = 0<\/em>, assim<\/p><figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"556\" height=\"38\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/image-12.png\" alt=\"\" class=\"wp-image-132046\" style=\"width:405px;height:auto\"><\/figure><h2 class=\"wp-block-heading\" id=\"h-estuda-com-a-coruja-para-suas-provas\"><span class=\"ez-toc-section\" id=\"Estuda-com-a-Coruja-para-suas-provas\"><\/span>Estuda com a Coruja para suas provas!<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Aqui no EV temos os conte&uacute;dos completos sobre o que mais aparece nos vestibulares, voc&ecirc; pode escolher o pacote que melhor atende suas demandas para o curso que deseja passar. Dessa <span id=\"urn:enhancement-97cbf45a-6048-465d-8505-f22ca28a72c7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span>, voc&ecirc; prioriza as &aacute;reas de estudo que mais precisa focar para fazer o curso dos seus sonhos. Para saber mais, acesse:<\/p><p class=\"wp-block-paragraph\"><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/cursos\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/EV2.jpg\" alt=\"CTA - Estrat\u00e9gia Vestibulares 2\" title=\"CTA - Estrat\u00e9gia Vestibulares 2\">\n        <\/a>\n\n\n\n\n<\/p><\/p>\n","protected":false},"excerpt":{"rendered":"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas\n","protected":false},"author":61,"featured_media":132014,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[],"class_list":["post-131999","post","type-post","status-publish","format-standard","has-post-thumbnail","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v28.2) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es<\/title>\n<meta name=\"description\" content=\"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas em vestibulares.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es\" \/>\n<meta property=\"og:description\" content=\"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas em vestibulares.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/\" \/>\n<meta property=\"og:site_name\" content=\"Estrat\u00e9gia Vestibulares\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/facebook.com\/estrategiavestibulares\" \/>\n<meta property=\"article:published_time\" content=\"2024-12-11T12:50:59+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-02-19T18:08:51+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/math.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1240\" \/>\n\t<meta property=\"og:image:height\" content=\"760\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Caiky Pinheiro\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@Estrategiavest\" \/>\n<meta name=\"twitter:site\" content=\"@Estrategiavest\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Caiky Pinheiro\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/\"},\"author\":{\"name\":\"Caiky Pinheiro\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#\\\/schema\\\/person\\\/c7f508351071581779a499fb483e23ed\"},\"headline\":\"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es\",\"datePublished\":\"2024-12-11T12:50:59+00:00\",\"dateModified\":\"2025-02-19T18:08:51+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/\"},\"wordCount\":1493,\"publisher\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/math.jpg\",\"articleSection\":[\"Matem\u00e1tica\"],\"inLanguage\":\"pt-BR\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/\",\"url\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/\",\"name\":\"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/math.jpg\",\"datePublished\":\"2024-12-11T12:50:59+00:00\",\"dateModified\":\"2025-02-19T18:08:51+00:00\",\"description\":\"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas em vestibulares.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#primaryimage\",\"url\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/math.jpg\",\"contentUrl\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/math.jpg\",\"width\":1240,\"height\":760,\"caption\":\"equa\u00e7\u00e3o-funcional-defini\u00e7\u00e3o-tipos-aplica\u00e7\u00f5es\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/equacao-funcional-definicao-tipos-e-aplicacoes\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Mat\u00e9rias\",\"item\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Matem\u00e1tica\",\"item\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/materias\\\/matematica\\\/\"},{\"@type\":\"ListItem\",\"position\":4,\"name\":\"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#website\",\"url\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/\",\"name\":\"Estrat\u00e9gia Vestibulares\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#organization\",\"name\":\"Estrat\u00e9gia Vestibulares\",\"url\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/wp-content\\\/uploads\\\/2021\\\/06\\\/Logo-Vestibulares@4x.png\",\"contentUrl\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/wp-content\\\/uploads\\\/2021\\\/06\\\/Logo-Vestibulares@4x.png\",\"width\":250,\"height\":66,\"caption\":\"Estrat\u00e9gia Vestibulares\"},\"image\":{\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/facebook.com\\\/estrategiavestibulares\",\"https:\\\/\\\/x.com\\\/Estrategiavest\",\"https:\\\/\\\/www.instagram.com\\\/estrategiavestibulares\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCGWdZy8ULI-7_5GrABa4vNA\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/#\\\/schema\\\/person\\\/c7f508351071581779a499fb483e23ed\",\"name\":\"Caiky Pinheiro\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/581b7703750b5199213f75080312fb21d4d9c345c3b6da1512bfca922df89180?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/581b7703750b5199213f75080312fb21d4d9c345c3b6da1512bfca922df89180?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/581b7703750b5199213f75080312fb21d4d9c345c3b6da1512bfca922df89180?s=96&d=mm&r=g\",\"caption\":\"Caiky Pinheiro\"},\"description\":\"Redator do Portal Estrat\u00e9gia Vestibulares e estudante de Matem\u00e1tica Aplicada na Unicamp.\",\"url\":\"https:\\\/\\\/vestibulares.estrategia.com\\\/portal\\\/author\\\/caiky-pinheiro\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es","description":"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas em vestibulares.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/","og_locale":"pt_BR","og_type":"article","og_title":"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es","og_description":"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas em vestibulares.","og_url":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/","og_site_name":"Estrat\u00e9gia Vestibulares","article_publisher":"https:\/\/facebook.com\/estrategiavestibulares","article_published_time":"2024-12-11T12:50:59+00:00","article_modified_time":"2025-02-19T18:08:51+00:00","og_image":[{"width":1240,"height":760,"url":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/math.jpg","type":"image\/jpeg"}],"author":"Caiky Pinheiro","twitter_card":"summary_large_image","twitter_creator":"@Estrategiavest","twitter_site":"@Estrategiavest","twitter_misc":{"Escrito por":"Caiky Pinheiro","Est. tempo de leitura":"7 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#article","isPartOf":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/"},"author":{"name":"Caiky Pinheiro","@id":"https:\/\/vestibulares.estrategia.com\/portal\/#\/schema\/person\/c7f508351071581779a499fb483e23ed"},"headline":"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es","datePublished":"2024-12-11T12:50:59+00:00","dateModified":"2025-02-19T18:08:51+00:00","mainEntityOfPage":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/"},"wordCount":1493,"publisher":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/#organization"},"image":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#primaryimage"},"thumbnailUrl":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/math.jpg","articleSection":["Matem\u00e1tica"],"inLanguage":"pt-BR"},{"@type":"WebPage","@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/","url":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/","name":"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es","isPartOf":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/#website"},"primaryImageOfPage":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#primaryimage"},"image":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#primaryimage"},"thumbnailUrl":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/math.jpg","datePublished":"2024-12-11T12:50:59+00:00","dateModified":"2025-02-19T18:08:51+00:00","description":"Entenda mais sobre o que s\u00e3o as equa\u00e7\u00f5es funcionais, qual a sua defini\u00e7\u00e3o, principais tipos e como elas s\u00e3o aplicadas em vestibulares.","breadcrumb":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/"]}]},{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#primaryimage","url":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/math.jpg","contentUrl":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/12\/math.jpg","width":1240,"height":760,"caption":"equa\u00e7\u00e3o-funcional-defini\u00e7\u00e3o-tipos-aplica\u00e7\u00f5es"},{"@type":"BreadcrumbList","@id":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/equacao-funcional-definicao-tipos-e-aplicacoes\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/vestibulares.estrategia.com\/portal\/"},{"@type":"ListItem","position":2,"name":"Mat\u00e9rias","item":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/"},{"@type":"ListItem","position":3,"name":"Matem\u00e1tica","item":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/"},{"@type":"ListItem","position":4,"name":"Equa\u00e7\u00e3o Funcional: defini\u00e7\u00e3o, tipos e aplica\u00e7\u00f5es"}]},{"@type":"WebSite","@id":"https:\/\/vestibulares.estrategia.com\/portal\/#website","url":"https:\/\/vestibulares.estrategia.com\/portal\/","name":"Estrat\u00e9gia Vestibulares","description":"","publisher":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/vestibulares.estrategia.com\/portal\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/vestibulares.estrategia.com\/portal\/#organization","name":"Estrat\u00e9gia Vestibulares","url":"https:\/\/vestibulares.estrategia.com\/portal\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/vestibulares.estrategia.com\/portal\/#\/schema\/logo\/image\/","url":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/06\/Logo-Vestibulares@4x.png","contentUrl":"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/06\/Logo-Vestibulares@4x.png","width":250,"height":66,"caption":"Estrat\u00e9gia Vestibulares"},"image":{"@id":"https:\/\/vestibulares.estrategia.com\/portal\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/facebook.com\/estrategiavestibulares","https:\/\/x.com\/Estrategiavest","https:\/\/www.instagram.com\/estrategiavestibulares","https:\/\/www.youtube.com\/channel\/UCGWdZy8ULI-7_5GrABa4vNA"]},{"@type":"Person","@id":"https:\/\/vestibulares.estrategia.com\/portal\/#\/schema\/person\/c7f508351071581779a499fb483e23ed","name":"Caiky Pinheiro","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/secure.gravatar.com\/avatar\/581b7703750b5199213f75080312fb21d4d9c345c3b6da1512bfca922df89180?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/581b7703750b5199213f75080312fb21d4d9c345c3b6da1512bfca922df89180?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/581b7703750b5199213f75080312fb21d4d9c345c3b6da1512bfca922df89180?s=96&d=mm&r=g","caption":"Caiky Pinheiro"},"description":"Redator do Portal Estrat\u00e9gia Vestibulares e estudante de Matem\u00e1tica Aplicada na Unicamp.","url":"https:\/\/vestibulares.estrategia.com\/portal\/author\/caiky-pinheiro\/"}]}},"_links":{"self":[{"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/posts\/131999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/users\/61"}],"replies":[{"embeddable":true,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/comments?post=131999"}],"version-history":[{"count":9,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/posts\/131999\/revisions"}],"predecessor-version":[{"id":135426,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/posts\/131999\/revisions\/135426"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/media\/132014"}],"wp:attachment":[{"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/media?parent=131999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/categories?post=131999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vestibulares.estrategia.com\/portal\/wp-json\/wp\/v2\/tags?post=131999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}