{"id":10,"date":"2023-09-17T11:13:44","date_gmt":"2023-09-17T11:13:44","guid":{"rendered":"https:\/\/mathority.org\/pt\/funcao-exponencial\/"},"modified":"2023-09-17T11:13:44","modified_gmt":"2023-09-17T11:13:44","slug":"funcao-exponencial","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/funcao-exponencial\/","title":{"rendered":"Fun\u00e7\u00e3o exponencial"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o fun\u00e7\u00f5es exponenciais e tamb\u00e9m como representar uma fun\u00e7\u00e3o exponencial em um gr\u00e1fico. Al\u00e9m disso, voc\u00ea ver\u00e1 todas as suas caracter\u00edsticas e diversos exemplos para entend\u00ea-lo totalmente. Por fim, voc\u00ea poder\u00e1 praticar exerc\u00edcios e problemas resolvidos passo a passo em fun\u00e7\u00f5es exponenciais. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-funcion-exponencial\"><\/span> O que \u00e9 uma fun\u00e7\u00e3o exponencial?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A defini\u00e7\u00e3o de uma fun\u00e7\u00e3o exponencial \u00e9 a seguinte: <\/p>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> Em matem\u00e1tica, <strong>fun\u00e7\u00f5es exponenciais<\/strong> s\u00e3o fun\u00e7\u00f5es que possuem a vari\u00e1vel independente <em>x<\/em> no expoente de uma pot\u00eancia. Em outras palavras, eles s\u00e3o os seguintes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73aa6cf687ce5e8c2f9faf41f5614e53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 um n\u00famero real positivo e diferente de 1. <\/p>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-funciones-exponenciales\"><\/span> Exemplos de fun\u00e7\u00f5es exponenciais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> As seguintes fun\u00e7\u00f5es s\u00e3o exemplos de fun\u00e7\u00f5es exponenciais: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf0f6145283a443b500a0bfebaec6ed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3^{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2ca442ee75f74bd911dd3ab296ac873_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=4^{-x}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2052342e5fc234eb4428c63b727b4e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\left( \\frac{1}{2} \\right)^x\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"105\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef2b8e1b29b1b10d376713f6fa34fdfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-las-funciones-exponenciales\"><\/span> Caracter\u00edsticas das fun\u00e7\u00f5es exponenciais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> As fun\u00e7\u00f5es exponenciais t\u00eam as seguintes propriedades:<\/p>\n<ul>\n<li> O dom\u00ednio de uma fun\u00e7\u00e3o exponencial consiste em n\u00fameros reais, ou em outras palavras, existe uma fun\u00e7\u00e3o exponencial para qualquer valor de <em>x<\/em> .<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91d461485d0f02bb14db6855a3774878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f=\\mathbff{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<ul>\n<li> No entanto, a fun\u00e7\u00e3o aceita apenas valores positivos, portanto o contradom\u00ednio de uma fun\u00e7\u00e3o exponencial consiste em n\u00fameros reais positivos.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36ed1f92f41441bd0c1d64faa0233a01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Im } f= (0,+\\infty)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Toda fun\u00e7\u00e3o exponencial \u00e9 uma fun\u00e7\u00e3o cont\u00ednua e injetiva.<\/li>\n<\/ul>\n<ul>\n<li> Se a fun\u00e7\u00e3o n\u00e3o for traduzida, qualquer fun\u00e7\u00e3o exponencial passa pelo ponto (0,1). Porque a fun\u00e7\u00e3o avaliada como zero sempre d\u00e1 um.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b49b6b1fa245db9037f372c69f2c1422_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=a^0=1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Da mesma forma, o valor de uma fun\u00e7\u00e3o exponencial em x=1 \u00e9 igual \u00e0 base.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2f27dba606789a982e3e3c373bda0d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=a^1=a\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Se a base de poder\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab31067f7ffbb9fa0478bfb791f6295f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"21\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 maior que 1, a fun\u00e7\u00e3o exponencial est\u00e1 aumentando. Por outro lado, se o coeficiente<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> est\u00e1 no intervalo entre 0 e 1, a fun\u00e7\u00e3o exponencial est\u00e1 diminuindo.<\/li>\n<\/ul>\n<ul>\n<li> Em geral, o eixo x \u00e9 uma ass\u00edntota horizontal de uma fun\u00e7\u00e3o exponencial.<\/li>\n<\/ul>\n<ul>\n<li> O inverso da fun\u00e7\u00e3o exponencial \u00e9 a fun\u00e7\u00e3o logar\u00edtmica. Portanto, os gr\u00e1ficos de uma fun\u00e7\u00e3o exponencial e de uma fun\u00e7\u00e3o logar\u00edtmica s\u00e3o sim\u00e9tricos em rela\u00e7\u00e3o \u00e0 reta y=x se ambas tiverem a mesma base. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-representar-en-una-grafica-una-funcion-exponencial\"><\/span> Como representar graficamente uma fun\u00e7\u00e3o exponencial<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> As fun\u00e7\u00f5es exponenciais s\u00e3o muito simples de representar. Ent\u00e3o, vamos ver como representar graficamente uma fun\u00e7\u00e3o exponencial em um gr\u00e1fico usando um exemplo.<\/p>\n<ul>\n<li> Trace a seguinte fun\u00e7\u00e3o exponencial em um gr\u00e1fico:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b460d8172e3f8e6633b62ab29e0f220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Nas fun\u00e7\u00f5es exponenciais n\u00e3o h\u00e1 necessidade de calcular o dom\u00ednio, pois sempre ser\u00e3o todos n\u00fameros reais:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cd1539b66edeb38040ed80168e1fd9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f = \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Basta, portanto, elaborar a tabela de valores. Como esses tipos de fun\u00e7\u00f5es mudam muito de um ponto para outro, calcularemos 5 pontos. Mas quanto mais pontos calcularmos, mais precisa ser\u00e1 a representa\u00e7\u00e3o da fun\u00e7\u00e3o. <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-201\">\n<div class=\"wp-block-column is-layout-flow\">\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e63b3712de818b714fafb781a7aed7b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)=2^0= 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"198\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-57601feaa5d5af3836989b041b20fd1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)=2^1= 2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"198\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c314f7ea92ec93e438d5ca2116be97fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2)=2^2= 4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"199\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce96f914e1abc10b13488bc390af9501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 \\ \\longrightarrow \\ f(-1)=2^{-1}= 0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"253\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a1526d8613884193ce46bf1d8515d8f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -2 \\ \\longrightarrow \\ f(-2)=2^{-2}= 0,25\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-299dcbadf5327b518849f641dd641f34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 0 &amp; 1 \\\\ 1 &amp; 2 \\\\ 2 &amp; 4 \\\\ -1 &amp; 0,5 \\\\ -2 &amp; 0,25 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"90\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Recomendamos o uso de uma calculadora para encontrar os pontos da tabela de valores, pois s\u00e3o complicados de calcular manualmente.<\/p>\n<p> Agora representamos os pontos em um gr\u00e1fico <strong>:<\/strong> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/cropped-polynomials-icon-1.png.png\" alt=\"\" class=\"wp-image-249\" width=\"274\" height=\"329\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E finalmente, juntamos os pontos e estendemos a fun\u00e7\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-representation-graphique-d-une-fonction-exponentielle.webp\" alt=\"como representar ou representar graficamente uma fun\u00e7\u00e3o exponencial\" class=\"wp-image-250\" width=\"274\" height=\"326\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Observe que a fun\u00e7\u00e3o \u00e0 direita continua a crescer at\u00e9 o infinito.<\/p>\n<p> Em contraste, a fun\u00e7\u00e3o \u00e0 esquerda diminui, mas nunca chega a 0. Mesmo chegando muito perto dela, nunca a toca. Isso significa que a reta y=0 (o eixo x) \u00e9 uma ass\u00edntota horizontal. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-funciones-exponenciales\"><\/span> Exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es exponenciais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Fa\u00e7a um gr\u00e1fico da seguinte fun\u00e7\u00e3o exponencial: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03ffe9b15e527e77fc38e7a090b9086d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 2^x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma fun\u00e7\u00e3o exponencial, portanto para represent\u00e1-la \u00e9 necess\u00e1rio criar uma tabela de valores dando valores \u00e0 vari\u00e1vel x: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-204\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c782f0c5fc2c9576431a0653e6e954da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)= 2^0+1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"228\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1a2424217eb051ccbc60bd9c9832cb16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)= 2^1+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"229\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46a35a2a78981d594a4599130a5128ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2)= 2^2+1=5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"228\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fceb9d30fc89075d6f6d367db21aefa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 \\ \\longrightarrow \\ f(-1)= 2^{-1}+1=1,5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"283\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1d391b6991faa22024aa6164e79fde1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -2 \\ \\longrightarrow \\ f(-2)= 2^{-2}+1=1,25\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"292\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5623433725def7676c75afcfdf23ad6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 0 &amp; 2 \\\\ 1 &amp; 3 \\\\ 2 &amp; 5 \\\\ -1 &amp; 1,5 \\\\ -2 &amp; 1,25 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"90\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Assim que tivermos a tabela de valores, plotamos os pontos obtidos no gr\u00e1fico e plotamos a fun\u00e7\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-etape-par-etape-des-fonctions-exponentielles.webp\" alt=\"exerc\u00edcio resolvido passo a passo de fun\u00e7\u00f5es exponenciais\" class=\"wp-image-251\" width=\"301\" height=\"298\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Observe que a fun\u00e7\u00e3o \u00e0 direita continua a crescer at\u00e9 o infinito. Por outro lado, \u00e0 esquerda, a fun\u00e7\u00e3o diminui, mas nunca excede 1. Na verdade, a fun\u00e7\u00e3o tem uma ass\u00edntota horizontal \u00e0 direita y=1.<\/p>\n<p class=\"has-text-align-left\"> Neste caso, a ass\u00edntota horizontal est\u00e1 em y=1 em vez do eixo OX porque uma transla\u00e7\u00e3o vertical de uma unidade para cima foi feita em dire\u00e7\u00e3o \u00e0 fun\u00e7\u00e3o.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 2<\/h3>\n<p> Trace a seguinte fun\u00e7\u00e3o exponencial em um gr\u00e1fico: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b9a894948fd1963ec5090a0e42d82da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left(\\frac{1}{3}\\right)^x\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"105\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma fun\u00e7\u00e3o exponencial, portanto para represent\u00e1-la graficamente \u00e9 necess\u00e1rio construir uma tabela de valores dando valores \u00e0 vari\u00e1vel x: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-207\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a1ce9dea9f80eeafd40e7e6e13b200ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)= \\left(\\cfrac{1}{3}\\right)^0 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"228\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-093a357ac3d5530f00f1f69ccd5a0ef4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)= \\left(\\cfrac{1}{3}\\right)^1 = 0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"255\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c324f0d99a2076cc1bfb2d38acb9b76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2)= \\left(\\cfrac{1}{3}\\right)^2 = 0,11\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"254\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e55ea725b4125b1f53dce7b5b42ea633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 \\ \\longrightarrow \\ f(-1)= \\left(\\cfrac{1}{3}\\right)^{-1} = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"267\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f1abd0ebf23e5d165e5d1cfd94afb4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -2 \\ \\longrightarrow \\ f(-2)= \\left(\\cfrac{1}{3}\\right)^{-2} = 9\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"267\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbf76eff59299da8ebaf3185eef2c654_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 0 &amp; 1 \\\\ 1 &amp; 0,33 \\\\ 2 &amp; 0,11 \\\\ -1 &amp; 3 \\\\ -2 &amp; 9 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"90\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Assim que tivermos a tabela de valores, plotamos os pontos calculados no gr\u00e1fico e desenhamos a fun\u00e7\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/polynomes-px.png\" alt=\"exerc\u00edcios de fun\u00e7\u00f5es exponenciais resolvidos\" class=\"wp-image-252\" width=\"266\" height=\"454\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Observe que a fun\u00e7\u00e3o \u00e0 esquerda continua a crescer at\u00e9 o infinito. Por outro lado, \u00e0 direita, a fun\u00e7\u00e3o diminui, mas nunca excede 0. Na verdade, a fun\u00e7\u00e3o tem uma ass\u00edntota horizontal em y=0 (o eixo X).<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> Trace a seguinte fun\u00e7\u00e3o exponencial em um gr\u00e1fico: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6abae87b33d734a0f0df3fda4b4af143_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left(\\frac{1}{2}\\right)^x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"136\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma fun\u00e7\u00e3o exponencial, portanto para desenh\u00e1-la \u00e9 necess\u00e1rio criar uma tabela de valores avaliando a fun\u00e7\u00e3o em v\u00e1rios pontos: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-210\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d86d4d4bb01fc8784c0e389848e82c3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 0 \\ \\longrightarrow \\ f(0)= \\left(\\cfrac{1}{2}\\right)^0+3 = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"259\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fa71d27c766686ff32712a8b1767e1ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 1 \\ \\longrightarrow \\ f(1)= \\left(\\cfrac{1}{2}\\right)^1+3 = 3,5\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"275\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-86b1fa3f8b4652a02cc02efc2a75bd80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2 \\ \\longrightarrow \\ f(2)= \\left(\\cfrac{1}{2}\\right)^2+3 = 3,25\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"284\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c27100fdc0eb9ede26c4704ed45d8914_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 \\ \\longrightarrow \\ f(-1)= \\left(\\cfrac{1}{2}\\right)^{-1}+3 = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"297\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7389ec7cd9ac887cb79f083ba84e92e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -2 \\ \\longrightarrow \\ f(-2)= \\left(\\cfrac{1}{2}\\right)^{-2}+3 = 7\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"298\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2d6c182e2785f23f50b3b16e4e183157_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; f(x) \\\\ \\hline 0 &amp; 4 \\\\ 1 &amp; 3,5 \\\\ 2 &amp; 3,25 \\\\ -1 &amp; 5 \\\\ -2 &amp; 7 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"90\" style=\"vertical-align: -62px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Por fim, representamos os pontos obtidos no gr\u00e1fico e tra\u00e7amos a fun\u00e7\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/cropped-polynomes-px.png.png\" alt=\"problema de fun\u00e7\u00e3o exponencial\" class=\"wp-image-253\" width=\"305\" height=\"372\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Observe que a fun\u00e7\u00e3o \u00e0 esquerda cresce ilimitadamente at\u00e9 o infinito. Por outro lado, \u00e0 direita, a fun\u00e7\u00e3o diminui mas nunca excede 3. Na verdade, a fun\u00e7\u00e3o tem uma ass\u00edntota horizontal em y=3.<\/p>\n<p class=\"has-text-align-left\"> Neste caso, a ass\u00edntota horizontal est\u00e1 em y=3 em vez do eixo X porque a fun\u00e7\u00e3o foi movida verticalmente tr\u00eas unidades para cima.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Resolva o seguinte problema relacionado a fun\u00e7\u00f5es exponenciais.<\/p>\n<ul>\n<li> Determine o valor de\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> de modo que a pr\u00f3xima fun\u00e7\u00e3o exponencial passe pelo ponto (2.8). <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be030f5f0ee0aada1e1c389763b331d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=k\\cdot 2^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o deve passar pelo ponto (2,8), ent\u00e3o podemos substituir os valores de <em>x<\/em> e <em>f(x)<\/em> do ponto na fun\u00e7\u00e3o para encontrar o valor da constante <em>k:<\/em><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bb3d3396e39826e0d6223a79bafe6bf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=k\\cdot 2^x \\ \\xrightarrow{x \\ = \\ 2 \\ ; \\ f(x) \\ = \\ 8} \\ 8 = k \\cdot 2^2\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"310\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora resolvemos a equa\u00e7\u00e3o resultante: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a345fdf8c61dc26e5092176b51a29e59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8 = k \\cdot 2^2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"71\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4683f83c268d651c548b98d2320441b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8 = k \\cdot 4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d793900a9bfa50bf89f48216a403e616_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{8}{4} = k\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"41\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-34ed53fa864ab9ac888742f196a6453c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{ 2 = k}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Resolva o seguinte problema relacionado a fun\u00e7\u00f5es exponenciais.<\/p>\n<p> Uma popula\u00e7\u00e3o de cupins se reproduz de acordo com a seguinte fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b41270bdb73979966e8c8e118e76cf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(t)=3^{t+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59d54ba73238a26eb8acae54cb83607e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(t)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"30\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u00e9 o n\u00famero de cupins e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> o tempo passou em meses.<\/p>\n<p> Quantos cupins existir\u00e3o ap\u00f3s 1 ano? <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para calcular o n\u00famero de cupins que haver\u00e1 em um ano, basta substituir o tempo decorrido (1 ano) na fun\u00e7\u00e3o. Mas como a fun\u00e7\u00e3o <em>t<\/em> s\u00e3o os meses decorridos e n\u00e3o os anos, devemos colocar <em>t<\/em> =12 j\u00e1 que num ano h\u00e1 12 meses: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b41270bdb73979966e8c8e118e76cf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(t)=3^{t+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68f69b244c701ac9238c9939e5d85791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(12)=3^{12+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-406c5d11e786f4624ffc7834f6b8fe8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(12)=3^{13}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"89\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos com a calculadora:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b44ece2cfd87c845152ec5aa481188fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(12)= 1594323\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, depois de um ano, haver\u00e1 1.594.323 cupins.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o fun\u00e7\u00f5es exponenciais e tamb\u00e9m como representar uma fun\u00e7\u00e3o exponencial em um gr\u00e1fico. Al\u00e9m disso, voc\u00ea ver\u00e1 todas as suas caracter\u00edsticas e diversos exemplos para entend\u00ea-lo totalmente. Por fim, voc\u00ea poder\u00e1 praticar exerc\u00edcios e problemas resolvidos passo a passo em fun\u00e7\u00f5es exponenciais. O que \u00e9 uma fun\u00e7\u00e3o exponencial? A &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/funcao-exponencial\/\"> <span class=\"screen-reader-text\">Fun\u00e7\u00e3o exponencial<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[22],"tags":[],"class_list":["post-10","post","type-post","status-publish","format-standard","hentry","category-representacao-de-funcao"],"yoast_head":"<!-- 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