{"id":60838,"date":"2022-11-09T11:10:11","date_gmt":"2022-11-09T14:10:11","guid":{"rendered":"https:\/\/militares.estrategia.com\/portal\/?p=60838"},"modified":"2024-09-30T10:59:32","modified_gmt":"2024-09-30T13:59:32","slug":"o-que-e-funcao","status":"publish","type":"post","link":"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/","title":{"rendered":"O que \u00e9 Fun\u00e7\u00e3o: dom\u00ednio, contradom\u00ednio e imagem"},"content":{"rendered":"\n<p>Se voc\u00ea tem o sonho de tornar-se um militar de carreira das For\u00e7as Armadas, sabe o n\u00edvel de dificuldade dos seletivo! Confira os resumos de Matem\u00e1tica que o Estrat\u00e9gia Militares preparou para voc\u00ea sobre<strong> o que \u00e9 Fun\u00e7\u00e3o e quais os seus tipos.&nbsp;&nbsp;&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Aten\u00e7\u00e3o!<\/strong> Para compreender melhor o assunto de Fun\u00e7\u00f5es, confira tamb\u00e9m os textos sobre <a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\" target=\"_blank\" >par ordenado, produto cartesiano<\/a> e<a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/\" target=\"_blank\" > rela\u00e7\u00f5es bin\u00e1rias<\/a>.&nbsp;<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_2 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\t<\/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:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#O-que-e-funcao\" >O que \u00e9 fun\u00e7\u00e3o?&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#Dominio-contradominio-e-imagem\" >Dom\u00ednio, contradom\u00ednio e imagem&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#Como-mostrar-a-funcao-em-um-grafico\" >Como mostrar a fun\u00e7\u00e3o em um gr\u00e1fico?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#Tipos-de-funcao\" >Tipos de fun\u00e7\u00e3o<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#Funcao-com-multiplas-variaveis\" >Fun\u00e7\u00e3o com m\u00faltiplas vari\u00e1veis<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#Na-pratica\" >Na pr\u00e1tica!&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao\/#Veja-tambem\" >Veja tamb\u00e9m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-e-funcao-nbsp\"><span class=\"ez-toc-section\" id=\"O-que-e-funcao\"><\/span>O que \u00e9 fun\u00e7\u00e3o?&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Uma fun\u00e7\u00e3o equivale a<strong> uma rela\u00e7\u00e3o bin\u00e1ria, s\u00f3 que com algumas restri\u00e7\u00f5es. <\/strong>Considere o exemplo da fun\u00e7\u00e3o \ud835\udc53: \ud835\udc34 \u2192 \ud835\udc35 que \u00e9 uma rela\u00e7\u00e3o bin\u00e1ria que relaciona elementos do conjunto A em elementos de B.\u00a0<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/g_bVZHuXKx7ecq5VysQ5EKwxNzViM3WxfONXL6ttKy2JxR28dq0rmi115pBCWLxyim97TTbz2hq4ie7bIu9Mb9aFU3IPh9FI5VZfv67210cEffZMeTyzns4bQ2TJkXduGhm9vg8ZB0BAXBCetJnFLLNJOBy3G8DThuzGgm_fCwXCDh5oGQl2j9mOOlK38g\" alt=\"\" style=\"width:320px;height:209px\"\/><\/figure><\/div>\n\n\n<p>No caso acima, os n\u00fameros {1, 2, 3} s\u00e3o relacionados \u00e0 sua paridade {\ud835\udc5d\ud835\udc4e\ud835\udc5f, \u00ed\ud835\udc5a\ud835\udc5d\ud835\udc4e\ud835\udc5f}. A defini\u00e7\u00e3o formal de fun\u00e7\u00e3o \u00e9 dada por:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/7Bux0TimpGS5h2zpn2sNZ9T4o_sRrgYf4wmyXtXumA7w9ROuzzXmxMEScE7BzEa5c6CmcUnyS_0_tEk252JIQEEyKqC6hTOZMqPL4jSYvxCEfbCxGOrgNID9vn1GCuclsvAE-zLKzQhyPYhxXX3UE55XTUA9RjzY5ImfWsoMD0fR_TmFI_BC2IImgwVFcg\" alt=\"\" style=\"width:-128px;height:-10px\"\/><\/figure><\/div>\n\n\n<p>Perceba que \ud835\udc53 transforma \ud835\udc65 \u2208 \ud835\udc34 em \ud835\udc66 \u2208 \ud835\udc35. A partir dessa defini\u00e7\u00e3o vemos que \u00e9 necess\u00e1rio satisfazer a <strong>duas condi\u00e7\u00f5es para uma rela\u00e7\u00e3o ser considerada uma fun\u00e7\u00e3o:<\/strong>&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Todo elemento \ud835\udc65 pertencente a \ud835\udc34 deve ser relacionado a algum elemento \ud835\udc66 pertencente a \ud835\udc35; e&nbsp;<\/li>\n\n\n\n<li>Um \ud835\udc65, elemento de \ud835\udc34, n\u00e3o pode se relacionar a mais de um \ud835\udc66, elemento de \ud835\udc35.&nbsp;<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/fWAsozQ0JqwCoV19MZOV-TbUKrfyfPmXG9kCeEUV_17tAlQUuUgSlpc8NFTGpX2qsuBMeiO0FYRZKRRPKsm3CsDWM9zFrkzN9jIe2atQh5GmblOH_AZdf4BX4cWmdt8Gm9kgioSR2iG9YxG0l8i6dBtdl9qgfuY61cWRV1NMHEiVIoaS0UC8IfFlf9n3VQ\" alt=\"\"\/><\/figure><\/div>\n\n\n<h3 class=\"wp-block-heading\" id=\"h-exemplos-de-funcao\">Exemplos de fun\u00e7\u00e3o<\/h3>\n\n\n\n<p>Confira mais algumas possibilidades de fun\u00e7\u00e3o para voc\u00ea se familiarizar com o conceito!&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/04seBpIAzLlbRs8x9A7fh6PlSR_q1HYvcOtcsIbRHH50LQSkZ2vnM5nIwnffgzJconb_1EfiCz8DFgBtjqMDxq2MR84-GmNT_cJJF1unXzYnLI23Uxr0gsVDgYfs4gHXYaegJSsSyeHsIoxdCvP7-36BCkO7UkN3_66sRDcsalIKU82AWKZ77MCJrvHyWA\" alt=\"\" style=\"width:-89px;height:-56px\"\/><\/figure><\/div>\n\n\n<p>No exemplo acima, as duas condi\u00e7\u00f5es da defini\u00e7\u00e3o s\u00e3o satisfeitas. Perceba que o fato de 4 e 5 estarem relacionados ao mesmo elemento \ud835\udc4f n\u00e3o contradiz nenhuma das condi\u00e7\u00f5es para ser fun\u00e7\u00e3o.&nbsp;<\/p>\n\n\n\n<p>Observe o caso de<strong> rela\u00e7\u00e3o bin\u00e1ria que n\u00e3o \u00e9 uma fun\u00e7\u00e3o:<\/strong>&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/gq29JlolwzYO_Ev5c4AudmIqFcWE2_JtDX6gCSvTShOiZ2ZsD1RAUBSnQLPcortL7hLa-epcE9CffQY9sOQmg7TMQxGHtCsDwLMnJT4L_Th35kQdHDg0tEzu1TTkTB4OirEorvmNur1vt91koyeM1ktd9tJLXwXXsHbxqHZ257K6jioWK9e1OTvbRigLwg\" alt=\"\" style=\"width:-23px;height:-16px\"\/><\/figure><\/div>\n\n\n<p>Como o elemento 5 do conjunto <em>A<\/em> n\u00e3o possui correspondente em <em>B<\/em>, a primeira condi\u00e7\u00e3o \u00e9 violada. Observe um segundo exemplo de rela\u00e7\u00e3o bin\u00e1ria que viola as condi\u00e7\u00f5es de fun\u00e7\u00e3o:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/Xbo9WtBF4nFZ_9gvXDbjdSV5hgrs8QiRbienXEPBIeLFj0dkWJXgr0rrWigxjnF-MMuPU-_F0PP8XVJAwCrn2zuKfMc_EFqOTRHQT8s0hOqTADyl89O2vZtcjNkF6fAGR1VwfH8cMFSuHtqf-SfEDnv1s1eWxsvsxHELQsNq9WMYLNF1e1mdpTh27s1VXw\" alt=\"\" style=\"width:-62px;height:-46px\"\/><\/figure><\/div>\n\n\n<p>No caso acima, a segunda condi\u00e7\u00e3o foi quebrada, j\u00e1 que o elemento 5 do conjunto <em>A <\/em>possui mais de um correspondente em <em>B<\/em>.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-dominio-contradominio-e-imagem-nbsp\"><span class=\"ez-toc-section\" id=\"Dominio-contradominio-e-imagem\"><\/span>Dom\u00ednio, contradom\u00ednio e imagem&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Observe a fun\u00e7\u00e3o \ud835\udc53: \ud835\udc34 \u2192 \ud835\udc35 representada pelo diagrama de flechas a seguir:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/gYh02Fi8eJwiACedz6Ass4FaTRHegaU65PSWngguF1tp4K7qbsCylB5AMlwgLjliy6PfWswaoE2RgN20t6-nA0DDnWIpOWuwnkKGu6HGHPWM8esFecLW-pCqc2A6mJJxCcWCCQmeUsEkzQER132U82hlkhJ_g1y-IfxPXtZMwWzgHXEjb9eymL_0kc-oKA\" alt=\"\" style=\"width:-684px;height:-449px\"\/><\/figure><\/div>\n\n\n<p>Chamamos de<strong> dom\u00ednio de uma fun\u00e7\u00e3o, o conjunto de onde as setas partem,<\/strong> no caso, o <em>A<\/em>. Assim:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc37<sub>\ud835\udc53<\/sub> = \ud835\udc34 = {1, 2, 3, 4, 5}<\/p>\n\n\n\n<p>J\u00e1 o <strong>contradom\u00ednio equivale ao conjunto <\/strong><strong><em>B<\/em><\/strong>, escrito como:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc36\ud835\udc37<sub>\ud835\udc53<\/sub> = \ud835\udc35 = {\ud835\udc4e, \ud835\udc4f, \ud835\udc50, \ud835\udc51, \ud835\udc52}<\/p>\n\n\n\n<p>A<strong> imagem de <\/strong><strong><em>f<\/em><\/strong><strong>, por sua vez, \u00e9 o conjunto formado pelos elementos \ud835\udc66 \u2208 \ud835\udc35, tal que \u2200\ud835\udc65 \u2208 \ud835\udc34<\/strong>, \ud835\udc53(\ud835\udc65) = \ud835\udc66. Ela pode ser escrita dessa forma:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc3c\ud835\udc5a<sub>\ud835\udc53<\/sub> = {\ud835\udc4e, \ud835\udc4f, \ud835\udc50}<\/p>\n\n\n\n<p>Veja que podemos representar a imagem de um conjunto por meio da fun\u00e7\u00e3o. Ou seja, \ud835\udc53(\ud835\udc34) \u00e9 a imagem da fun\u00e7\u00e3o \ud835\udc53. J\u00e1 que:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc34) = \ud835\udc3c\ud835\udc5a\ud835\udc53 = {\ud835\udc4e, \ud835\udc4f, \ud835\udc50}<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-como-mostrar-a-funcao-em-um-grafico\"><span class=\"ez-toc-section\" id=\"Como-mostrar-a-funcao-em-um-grafico\"><\/span>Como mostrar a fun\u00e7\u00e3o em um gr\u00e1fico?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Assim como as rela\u00e7\u00f5es bin\u00e1rias, <strong>a fun\u00e7\u00e3o pode ser representada graficamente em um plano cartesiano. <\/strong>Para esbo\u00e7ar seu gr\u00e1fico, consideramos que \ud835\udc53 gera pares ordenados da forma (\ud835\udc65, \ud835\udc66) tal que \ud835\udc66 = \ud835\udc53(\ud835\udc65), com \ud835\udc65 \u2208 \ud835\udc34 e \ud835\udc66 \u2208 \u211d.<\/p>\n\n\n\n<p>Veja o exemplo a seguir, em que a rela\u00e7\u00e3o \ud835\udc53 de \ud835\udc34 em \u211d, com \ud835\udc34 = [0, 3], \u00e9 uma fun\u00e7\u00e3o:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/VsxLS3x1Jego09qxMSQhNrEhID0rVMbyB_Vy0CVD9fUm-W0JUXwL1EOumBxiA-aFe1MpkNmbgbK9j3ML_5EiePzvqiV-UDMSlDXmlT_Dw6na5veq1QdT4lhAYGxnkK95l67f_KXVy5UM6akY2LZTtp4Hjmbkc7Eo_blWNp1VnyWOP8oQbuJjpwbmBO-aRw\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>As retas verticais que passam pela rela\u00e7\u00e3o \ud835\udc53 <strong>possuem apenas um ponto de encontro<\/strong> na rela\u00e7\u00e3o, destacado em azul &#8211; o que satisfaz \u00e0s condi\u00e7\u00f5es da defini\u00e7\u00e3o de fun\u00e7\u00e3o. Assim, os elementos de <em>A<\/em> = [0,3] representados no eixo \ud835\udc65 correspondem a um \u00fanico valor em \u211d, representado no eixo \ud835\udc66.&nbsp;<\/p>\n\n\n\n<p>Agora, veja uma representa\u00e7\u00e3o gr\u00e1fica da rela\u00e7\u00e3o \ud835\udc53 de \ud835\udc34 em \u211d, com \ud835\udc34 = [1, 3] que n\u00e3o se refere a uma fun\u00e7\u00e3o:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/I_vKlrq35FHbAvoG4tH-v26cGgREOnz92NI1EXO6R2jbcMpXSw6fmEhbdIV_0cs-kUHA7lmlLVw1TsGCw2uIJgZpGPVoZYxnAX4tmHr4zssnJGUgvwOtksaSHM0XoItmf1L2IUUBF_Rg5rsp4VM4VlbL11bxoNvVNAuZH9RPLrQ4eD_l1fjC1DIaF_g39g\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Como h\u00e1 retas verticais que encontram o gr\u00e1fico de \ud835\udc53 em 2 pontos, <strong>ocorre uma viola\u00e7\u00e3o da condi\u00e7\u00e3o de \ud835\udc65 \u2208 \ud835\udc34 possuir apenas um correspondente em \ud835\udc66 \u2208 \u211d.<\/strong> Assim, para verificarmos se uma rela\u00e7\u00e3o \ud835\udc53 de \ud835\udc34 em \ud835\udc35 \u00e9 fun\u00e7\u00e3o por meio da representa\u00e7\u00e3o cartesiana, devemos ver se as retas paralelas ao eixo \ud835\udc66 encontram o gr\u00e1fico de \ud835\udc53 em um s\u00f3 ponto.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-tipos-de-funcao\"><span class=\"ez-toc-section\" id=\"Tipos-de-funcao\"><\/span>Tipos de fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Existem v\u00e1rios tipos e, por isso, separamos aquelas que t\u00eam<strong> maior possibilidade de serem cobradas na sua prova!&nbsp;<\/strong><\/p>\n\n\n\n<p>1) Fun\u00e7\u00e3o Afim<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \u211d \u2192 \u211d e \ud835\udc53(\ud835\udc65) = \ud835\udc4e\ud835\udc65 + \ud835\udc4f, \ud835\udc4e, \ud835\udc4f \u2208 \u211d<\/p>\n\n\n\n<p>2) Fun\u00e7\u00e3o Racional&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/xTc781uUs8m3hhm-C89OlIBh7riPRqU8zBorGUKot45xtJSgIQwRWypjEkAdiWtVZ9wYK26aG1pf93T_7T_DHnWpxSyexJlNa84eTp1fYc2-Ycyct6Ko_2P3lbYgIOVfU_b7EGPvIxA4M5TXhQvdovO-EL1tCyMNXfrlcbKs5nwlezIi2hB4sGCzkh9ysA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>3) Fun\u00e7\u00e3o Quadr\u00e1tica<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \u211d \u2192 \u211d e \ud835\udc53(\ud835\udc65) = \ud835\udc4e\ud835\udc65<sup>2<\/sup> + \ud835\udc4f\ud835\udc65 + \ud835\udc50; \ud835\udc4e \u2260 0<\/p>\n\n\n\n<p>4) Fun\u00e7\u00e3o Modular<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \u211d \u2192 \u211d e \ud835\udc53(\ud835\udc65) = |\ud835\udc65|<\/p>\n\n\n\n<p>5) Fun\u00e7\u00e3o Exponencial<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \u211d \u2192 \u211d e \ud835\udc53(\ud835\udc65) = \ud835\udc4e<sup>\ud835\udc65<\/sup> ; \ud835\udc4e &gt; 0 e \ud835\udc4e \u2260 1<\/p>\n\n\n\n<p>6) Fun\u00e7\u00e3o Logar\u00edtmica<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \u211d<sup>\u2217<\/sup><sub>+<\/sub>\u2192 \u211d e \ud835\udc53(\ud835\udc65) = log<sub>\ud835\udc4e<\/sub> \ud835\udc65 ; \ud835\udc4e &gt; 0 e \ud835\udc4e \u2260 1<\/p>\n\n\n\n<p>7) Fun\u00e7\u00e3o M\u00e1ximo Inteiro<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \u211d \u2192 \u211d e \ud835\udc53(\ud835\udc65) = \u230a\ud835\udc65\u230b<\/p>\n\n\n\n<p>8) Fun\u00e7\u00e3o Trigonom\u00e9trica<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53: \ud835\udc34 \u2192 \ud835\udc35 e \ud835\udc53(\ud835\udc65) = \ud835\udc60\ud835\udc52\ud835\udc5b\ud835\udc65 \ud835\udc5c\ud835\udc62 \ud835\udc53(\ud835\udc65) = \ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65 \ud835\udc5c\ud835\udc62 \ud835\udc53(\ud835\udc65) = \ud835\udc61\ud835\udc54\ud835\udc65<\/p>\n\n\n\n<p>9) Cosseno hiperb\u00f3lico em \ud835\udc65<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/nastvqYxVHYkePVszdPG6EcfmjyLIXME-8ofAxK9pJPSTuS21oFQZsY1rPw7spg6JAPjcZ02XIg2btnz_QlH3bSsTavy-DJYCdNJMrwx-T5ZT6Od4BzlTmK_dukU_-tLgsARkuAYUoFH_ewGSUxXo7N_1oCVTffx2Qnvz_mQbiCx3Oo4kTQr1K4tWjNefA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>10) Seno hiperb\u00f3lico em \ud835\udc65<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/z7TOK5MKIQuHQ9y_ZqlB7JKE9K5pP7JpsGZ484-VXYRZ9Ux44ZmBA6GrN_GvljTNP7UX7hDdmtGsKcO6kNZscEddyrwhkDN_F6kV9W1vayROkPEOFj8ZcnSFxKa92fSe7DweVnrFRkjDrFt6y1OHEzbD9RrC3VyYnV2zni114_EijgxOJRc8vGg9JK56YA\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>11) Tangente hiperb\u00f3lico em \ud835\udc65<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/g0UdJif72NncyQc2yKZAWOmE1T2gvOU9FLyi7McYp8UUyLFPeHUrC8v-w8Zdjf9cxcVp73ctFrU0fXNFXKBfRjrOHhMEAtRZxrIv1lAHn7GZaRhGkn_JL0LxojkLVOqC7uDqbA28deVkIoqQOGQPrK8R7-iTwrBfO1TQt5g3duHFGysBXRczqOAt_7qArw\" alt=\"\"\/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"h-funcao-com-multiplas-variaveis\"><span class=\"ez-toc-section\" id=\"Funcao-com-multiplas-variaveis\"><\/span>Fun\u00e7\u00e3o com m\u00faltiplas vari\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Voc\u00ea sabia que as <strong>fun\u00e7\u00f5es tamb\u00e9m podem ter mais de uma vari\u00e1vel<\/strong>? Veja os exemplos a seguir!<\/p>\n\n\n\n<p><strong>I) \ud835\udc53: \u211d<sup>2<\/sup> \u2192 \u211d<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65; \ud835\udc66) = \ud835\udc65<sup>2<\/sup> + \ud835\udc66<sup>2<\/sup><\/p>\n\n\n\n<p>Nesse caso, a fun\u00e7\u00e3o apresenta duas vari\u00e1veis, \ud835\udc65 e \ud835\udc66. Se atribuirmos \ud835\udc65 = 1 e \ud835\udc66 = 2, ela retornar\u00e1:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(1; 2) = 1<sup>2<\/sup> + 2<sup>2<\/sup> = 1 + 4 \u21d2 \ud835\udc53(1; 2) = 5<\/p>\n\n\n\n<p><strong>II) \ud835\udc53: \u211d<sup>3<\/sup> \u2192 \u211d<sup>2<\/sup><\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(\ud835\udc65; \ud835\udc66; \ud835\udc67) = (2\ud835\udc65 \u2212 \ud835\udc66 + \ud835\udc67; \ud835\udc65 + \ud835\udc66 \u2212 \ud835\udc67)<\/p>\n\n\n\n<p>O resultado da fun\u00e7\u00e3o com tr\u00eas vari\u00e1veis acima \u00e9 o par ordenado (2\ud835\udc65 \u2212 \ud835\udc66 + \ud835\udc67; \ud835\udc65 + \ud835\udc66 \u2212 \ud835\udc67). Assim, se atribuirmos \ud835\udc65 = 1, \ud835\udc66 = 2 e \ud835\udc67 = 3, temos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc53(1; 2; 3) = (2 \u22c5 1 \u2212 2 + 3; 1 + 2 \u2212 3) \u21d2 \ud835\udc53(1; 2; 3) = (3; 0)<\/p>\n\n\n\n<p><strong>III) \ud835\udc53: \u211d<sup>\ud835\udc5b<\/sup> \u2192 \u211d<sup>\ud835\udc5a<\/sup><\/strong>. Nesse exemplo, a fun\u00e7\u00e3o transforma n-uplas ordenadas em m-uplas ordenadas.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-na-pratica-nbsp\"><span class=\"ez-toc-section\" id=\"Na-pratica\"><\/span>Na pr\u00e1tica!&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Agora que voc\u00ea sabe o que \u00e9 uma fun\u00e7\u00e3o, que tal testar seus conhecimentos?<strong> Resolva o exerc\u00edcio a seguir!&nbsp;<\/strong><\/p>\n\n\n\n<p><em>Classifique as rela\u00e7\u00f5es abaixo, <\/em><strong><em>indicando se s\u00e3o ou n\u00e3o fun\u00e7\u00e3o<\/em><\/strong><em>. Caso positivo, <\/em><strong><em>determine seu dom\u00ednio, contradom\u00ednio e imagem.<\/em><\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/zRm7y3Nv3798a9KRo1kwzsNWckIUVSwaS-8UbWe5WRgc8Gd6TCilInSmvbsMn9Qrcy6UKZrwH-RNplL3jzCErbMxguJ4n-ju8sAy_FJug4lTxutldabz058LUpLGcTDToSgabhcHt4HXKIG4dyjfTUPM2M9Cbi2iGsjRai7jLXomaCAijhPkCwBCczZV3A\" width=\"266\" height=\"221\"><em><\/em><\/li>\n\n\n\n<li><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/bUPORzekDH41ZwUukyH5BbXRr30DCaHR9HMXDCR7VSB8f392ZKNs_z8rU1stXLOJhCK6KfsY5r3aKMukw_kdcC-qZKcU2UjdJJP6ekNCIJtdnrpHyerRQ4dVNIkXugLMeyCQnWc2vDY929BPbWYA_yMMmgkap1EWzjcwEyZd5Z7m4goO1L1NVFy1Q-ebLg\" width=\"284\" height=\"239\"><em><\/em><\/li>\n\n\n\n<li><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/rMH0Uu7Si0EHX7cZvQCKY3WPSsh0DAbYpfEFeYToDb-EfwnEllCfuPLgC4_vS0i7_rr5SfGtl-U5rtGpe5blxmnqFqZZvkvPAwR1f06WBR_MN92RuE5WW38cwh9jg8t-r1OT_jrEU9C-TR9miilQ_sq_9xJ38mSMjqCYZCkr8SHOb8lR9HINRqw7CLQ9Iw\" width=\"282\" height=\"238\"><em><\/em><\/li>\n\n\n\n<li><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/zb8BwPw9ffVGZ9PkjM91Qwbr0ESxA0V27va_VP1xCQVq-0ojn2AbRAnUQaC3LB3LEMZK8MYSMvJtgX0ZzI1vEWPW_YLES0oKmJELGEJxp7kc8u3G0IG23ERRLwFCVv0FuI3tZ-RVnMM6a-Hf1-Bmmb99OHqiAO4e799C5kbHduVQPjnQG0w_PQqvEw748g\" width=\"289\" height=\"245\"><em><\/em><\/li>\n<\/ol>\n\n\n\n<p><strong>Resolu\u00e7\u00e3o:&nbsp;<\/strong><\/p>\n\n\n\n<p>a) A rela\u00e7\u00e3o \ud835\udc45 n\u00e3o \u00e9 fun\u00e7\u00e3o, pois o elemento \u22121 possui dois correspondentes (0 e 3).<\/p>\n\n\n\n<p>b) A rela\u00e7\u00e3o \ud835\udc47 \u00e9 fun\u00e7\u00e3o. Todos os elementos do conjunto \ud835\udc34 possuem correspondentes em \ud835\udc35 e cada um deles \u00e9 associado a um \u00fanico elemento de \ud835\udc35. O dom\u00ednio de \ud835\udc47 \u00e9 o conjunto \ud835\udc34:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc37<sub>\ud835\udc47<\/sub> = {\u22121, 0, 1}<\/p>\n\n\n\n<p>O contradom\u00ednio \u00e9 o conjunto \ud835\udc35:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc36\ud835\udc37<sub>\ud835\udc47<\/sub> = {0, 1, 2, 3, 4, 5, 6}<\/p>\n\n\n\n<p>A imagem de \ud835\udc47 \u00e9 composta por todos os elementos de \ud835\udc35 que s\u00e3o associados a algum elemento de \ud835\udc34:<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc3c\ud835\udc5a<sub>\ud835\udc47<\/sub> = {0, 2, 6}<\/p>\n\n\n\n<p>c) A rela\u00e7\u00e3o \ud835\udc46 \u00e9 fun\u00e7\u00e3o. Todos elementos de \ud835\udc34 correspondem a algum elemento de \ud835\udc35 e cada um deles possui apenas um elemento associado a eles.<\/p>\n\n\n\n<p class=\"has-text-align-center\">\ud835\udc37<sub>\ud835\udc46<\/sub> = {\u22121, 0, 1}<br>\ud835\udc36\ud835\udc37<sub>\ud835\udc46<\/sub> = {0, 1, 2, 3, 4, 5, 6}<br>\ud835\udc3c\ud835\udc5a<sub>\ud835\udc46<\/sub> = {2, 4}<\/p>\n\n\n\n<p>d) A rela\u00e7\u00e3o \ud835\udc49 n\u00e3o \u00e9 fun\u00e7\u00e3o, pois o elemento 1 do conjunto \ud835\udc34 n\u00e3o corresponde a nenhum elemento em \ud835\udc35.<\/p>\n\n\n\n<p>E a\u00ed, guerreiro? Se quiser praticar mais, conhe\u00e7a o Banco de Quest\u00f5es do Estrat\u00e9gia Militares! S\u00e3o milhares de quest\u00f5es in\u00e9ditas e de prova anteriores para voc\u00ea treinar o que aprendeu! Clique no banner e conhe\u00e7a a nossa plataforma:&nbsp;<\/p>\n\n\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb\u00e9m:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-pa\/\">Progress\u00e3o Aritm\u00e9tica (PA): o que \u00e9, tipos e propriedades<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/\">Progress\u00e3o Geom\u00e9trica (PG): o que \u00e9, propriedades e mais<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\">Par ordenado e produto cartesiano: o que s\u00e3o e suas propriedades<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/\">O que \u00e9 uma Rela\u00e7\u00e3o Bin\u00e1ria na Matem\u00e1tica<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entenda-a-equacao-geral-da-reta-na-geometria-analitica\/\">Entenda a equa\u00e7\u00e3o geral da reta na Geometria Anal\u00edtica<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/estatistica-descritiva-conheca-tecnicas-e-aplicacoes\/\">Estat\u00edstica descritiva: conhe\u00e7a t\u00e9cnicas e aplica\u00e7\u00f5es<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/polinomios-definicoes-operacoes-fatoracao-e-mais\/\">Polin\u00f4mios: defini\u00e7\u00f5es, opera\u00e7\u00f5es, fatora\u00e7\u00e3o e mais!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/conheca-os-poligonos-regulares-e-suas-propriedades\/\">Conhe\u00e7a os pol\u00edgonos regulares e suas propriedades!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/matriz-entenda-os-conceitos-basicos\/\">Matriz: entenda os conceitos b\u00e1sicos!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/regra-de-tres-domine-facilmente-essa-ferramenta\/\">Regra de tr\u00eas: domine facilmente essa ferramenta!<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/entendendo-os-numeros-primos-e-sua-importancia-na-matematica\/\">Entendendo os n\u00fameros primos e sua import\u00e2ncia na matem\u00e1tica<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/inequacao-e-funcao-de-1o-grau-como-resolver\/\">Inequa\u00e7\u00e3o e Fun\u00e7\u00e3o de 1\u00ba grau: como resolver?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-funcao-afim\/\">O que \u00e9 Fun\u00e7\u00e3o Afim e quais suas propriedades?<\/a><\/li>\n<\/ul>\n\n\n\n<p><em>Conte\u00fado escrito com base no material elaborado pelo prof. Victor So.<\/em><\/p>\n\n\n","protected":false},"excerpt":{"rendered":"Se voc\u00ea tem o sonho de tornar-se um militar de carreira das For\u00e7as Armadas, sabe o n\u00edvel de&hellip;\n","protected":false},"author":9,"featured_media":60839,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[307],"tags":[],"wl_entity_type":[187],"class_list":{"0":"post-60838","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>O que \u00e9 Fun\u00e7\u00e3o: dom\u00ednio, contradom\u00ednio e imagem<\/title>\n<meta name=\"description\" content=\"Entenda o que \u00e9 Fun\u00e7\u00e3o, seus tipos e conceitos como dom\u00ednio, 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