{"id":33758,"date":"2021-06-26T13:00:26","date_gmt":"2021-06-26T13:00:26","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33758"},"modified":"2025-07-30T22:58:12","modified_gmt":"2025-07-30T22:58:12","slug":"domainium-ambitus-et-graphicum-functionum-algebraicarum","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/domainium-ambitus-et-graphicum-functionum-algebraicarum\/","title":{"rendered":"Domainium, Ambitus et Graphicum Functionum Algebraicarum"},"content":{"rendered":"<p><center><\/p>\n<h1>Domainium, Ambitus et Graphicum Functionum Algebraicarum<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nHaec lectio introducit notiones domainii, ambitus et graphicorum functionum, applicando eas ad exempla practica functionum algebraicarum. Examinantur technicae graphicae et analyticae ad hos elementa determinanda.<br \/>\n<\/em><br \/>\n<strong>Proposita Discendi:<\/strong><br \/>\nPost hanc lectionem discipulus poterit:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Definire<\/strong> recte domainium, ambitum et graphium functionis.<\/li>\n<li><strong>Adhibere<\/strong> methodos graphicas ad determinandum domainium et ambitum functionum algebraicarum.<\/li>\n<li><strong>Construere<\/strong> tabulas signorum ad analysandum comportamentum functionum.<\/li>\n<\/ol>\n<p><\/center><\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/zhb8GKlcdA8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n<h2>Definitio domainii, ambitus et graphii<\/h2>\n<p style=\"text-align: justify;\">Iam ad hoc punctum studium accuratum de functionibus linearibus, quadraticis et similibus perfecimus. Examinavimus etiam curvas sicut rectas, parabolas, ellipses et hyperbolas necnon operationes cum polynomis et functionibus algebraicis in genere. His peractis, nunc facilius erit penetrare aspectus aliquantum fundamentales circa functiones in genere, quod nunc incipiemus per introductionem notionum <strong>domainii, ambitus et graphii.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=306s\" target=\"_blank\" rel=\"noopener\"><strong>Sit <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> functio<\/strong><\/a> definita inter coniunctiones <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{matrix}f &amp; : &amp; A &amp; \\longrightarrow &amp; B \\\\ &amp; &amp; x &amp; \\longmapsto &amp; y=f(x)\n\n\\end{matrix}<\/span>\n<p style=\"text-align: justify;\">Coniunctiones <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> dicuntur esse coniunctiones \u00abinput\u00bb et \u00aboutput\u00bb, respective. Ex his definiuntur sequentia coniunctionum:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Dom(f) = \\{x\\in A\\;|\\; (\\exists y \\in B)(y=f(x))\\}<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Rec(f) = \\{y\\in B\\;|\\; (\\exists ! x \\in Dom(f))(y=f(x))\\}<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Graf(f) = \\{(x,y)\\in A\\times B\\;|\\; x\\in Dom(f) \\wedge y=f(x) \\}<\/span>\n<h2>Exemplorum Analysis<\/h2>\n<p style=\"text-align: justify;\">Omnia quae discere possumus de notionibus domainii, ambitus et graphii, quamvis essentia theoretica sint, intellectus eorum magis in evolutione exemplorum practicatorum consistit, quod nunc faciemus per analysin sequentium trium casuum:<\/p>\n<h3>Computare domainium, ambitum et graphium huius functionis: <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\sqrt{1-x^2}<\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=560s\" target=\"_blank\" rel=\"noopener\"><strong>Incipiamus hanc analysin<\/strong><\/a> scribendo <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x).<\/span> Si hoc facimus, tum obtinebimus aequationem<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y = \\sqrt{1-x^2}<\/span>\n<p style=\"text-align: justify;\">Si hanc expressionem ad quadratum elevamus, cito ad expressionem perveniemus quae ad quaestiones iam notas nos ducit<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; y^2 = 1-x^2 \\\\\n\n\\equiv &amp; x^2 + y^2 = 1 \\end{array}<\/span>\n<p style=\"text-align: justify;\">Haec est aequatio circuli unitarii.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-DQGthMyBY6g\/YNVVrnVQEfI\/AAAAAAAAFOQ\/6_lf8fRQdDIT9NMqstyLOJ2F7nQM9pc8ACLcBGAsYHQ\/s0\/circulounitario.PNG\" alt=\"Circulus Unitarius et domainium, ambitus et graphium\" class=\"aligncenter lazyload\" width=\"245\" height=\"249\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-DQGthMyBY6g\/YNVVrnVQEfI\/AAAAAAAAFOQ\/6_lf8fRQdDIT9NMqstyLOJ2F7nQM9pc8ACLcBGAsYHQ\/s0\/circulounitario.PNG\" alt=\"Circulus Unitarius et domainium, ambitus et graphium\" class=\"aligncenter lazyload\" width=\"245\" height=\"249\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Attamen, hic cauti esse debemus, quia elevando ad quadratum \u00abinformationem addidimus\u00bb. Algebraice duo valores exstant qui conditionem \u00abesse radicem quadratam\u00bb satisfaciunt, sed in initio huius analysis radix ut functio determinata est, et functiones tantum unum exitum admittere possunt. Agitur de radice principali. Hac de causa, propositum originale tantum ad partem superiorem circuli spectat, non ad totam figuram.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-AxSf-9lgnuE\/YNVbSJpd-rI\/AAAAAAAAFOg\/0APXEMWIFpAm8DX9651iD6wcq5bTJwFoQCLcBGAsYHQ\/s0\/circulounitario%2B2.PNG\" alt=\"circulus unitarius et domainium, ambitus et graphium\" class=\" aligncenter lazyload\" width=\"401\" height=\"361\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-AxSf-9lgnuE\/YNVbSJpd-rI\/AAAAAAAAFOg\/0APXEMWIFpAm8DX9651iD6wcq5bTJwFoQCLcBGAsYHQ\/s0\/circulounitario%2B2.PNG\" alt=\"circulus unitarius et domainium, ambitus et graphium\" class=\" aligncenter lazyload\" width=\"401\" height=\"361\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Ex hac figura manifestum est:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Dom(f) = \\{x\\in\\mathbb{R}\\;|\\; |x|\\leq 1\\} = [-1,1]<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Rec(f) = \\{y\\in\\mathbb{R}\\;|\\; 0\\leq y\\leq 1\\} = [0,1]<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Graf(f) = \\{(x,y)\\in \\mathbb{R}\\times \\mathbb{R}\\;|\\; x\\in [-1,1] \\wedge y=\\sqrt{1-x^2}\\}<\/span>\n<p style=\"text-align: justify;\">Quamvis hanc analysin ex perspectiva graphica elaboraverim, fieri quoque potest hoc ex approbatione magis analytica, sufficit operationes implicatas examinare.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\color{red}{\\sqrt{{1-x^2}}}<\/span>\n<p style=\"text-align: justify;\">Pars <span class=\"katex-eq\" data-katex-display=\"false\">1-x^2<\/span> bene definita est pro omnibus numeris realibus<\/p>\n<p style=\"text-align: justify;\">Contra, radix quadrata tantum valores maiores aut aequales zero admittit<\/p>\n<p style=\"text-align: justify;\">Ex hoc habetur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rlrl}\n\nx\\in Dom(f) &amp; \\leftrightarrow &amp; 0 &amp;\\leq 1-x^2 \\\\\n\n{} &amp; \\leftrightarrow &amp; x^2 &amp;\\leq 1 \\\\\n\n&amp; \\leftrightarrow &amp; |x| &amp;\\leq 1 \\\\\n\n&amp; \\leftrightarrow &amp; -1 &amp;\\leq x \\leq 1 \\\\\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nErgo:\\; Dom(f) = \\{x\\in \\mathbb{R}\\;|x| \\leq 1\\} = [-1,1]\n<\/span>\n<p style=\"text-align: justify;\">Methodi analytici ad ambitum determinandum plerumque difficiliores sunt; simplicissimae causae solvuntur per inventionem functionis inversae, sed antequam hanc rem plene investigemus, utile est compositionem functionum et alias causas simpliciores explorare ad firmam fundamentum constituendum. Interim, methodi graphici quos mox tractabimus, maiorem partem difficultatum circa ambitum determinandum obtegunt.<\/p>\n<h3>Analysis pro: <span class=\"katex-eq\" data-katex-display=\"false\">g(x) =\\displaystyle \\frac{x^2 - 1}{x^2 + 1}<\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong>Modus ad domainium<\/strong><\/a> celeriter inveniendum est interrogare de valoribus <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> qui \u00abfunctionem corrumpunt\u00bb. Manifestum est functionem corrumpi tantum si denominator ad nihilum redigitur. Id est:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n&amp; x^2 + 1 = 0 \\\\\n\n\\equiv &amp; x^2 = -1 \\\\\n\n\\end{array}<\/span>\n<p style=\"text-align: justify;\">Cum nullus numerus realis talem conditionem satisfacere possit, clarum est ergo<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\color{blue}{Dom(g) = \\mathbb{R}}<\/span>\n<p style=\"text-align: justify;\">Graphium determinare est plerumque via celerior ad ambitum functionis cognoscendum; ad hoc efficiendum, <a href=\"https:\/\/toposuranos.com\/algebra-de-polinomios-de-numeros-reales\/\" rel=\"noopener\" target=\"_blank\">divisio polynomiorum<\/a> erit instrumentum utile.<\/p>\n<p style=\"text-align: justify;\">Per divisionem polynomiorum obtinebimus:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle\\frac{x^2-1}{x^2+1} =<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">-\\displaystyle\\frac{2}{x^2 + 1}<\/span>\n<p style=\"text-align: justify;\">Ita, functionem originalem in duas partes simpliciores separavimus, quas \u00abpartem integram\u00bb et \u00abpartem fractionariam\u00bb vocamus. Has partes separatim graphice repraesentare multo facilius est quam totam functionem uno ictu describere.<\/p>\n<h3>Analysis pro: <span class=\"katex-eq\" data-katex-display=\"false\">h(x) =\\displaystyle \\frac{x - 1}{\\sqrt{x+1}}<\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=1580s\" target=\"_blank\" rel=\"noopener\"><strong>Analysis algebraica<\/strong><\/a> iuvabit ad domainium huius functionis cito determinandum. Sufficit animadvertere functionem bene definitam fore quoties<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rrl}\n\n&amp; 0 &amp; \\lt x + 1 \\\\\n\n\\equiv &amp; -1 &amp; \\lt x \\\\\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify;\">Ergo patet <span class=\"katex-eq\" data-katex-display=\"false\">Dom(h)=]-1,+\\infty[.<\/span>\n<p style=\"text-align: justify;\">Ad ambitum inveniendum expedit graphium delineare, quod simpliciter fieri potest utens <strong>tabula signorum.<\/strong> Functio <span class=\"katex-eq\" data-katex-display=\"false\">h(x)<\/span> componitur ex duabus partibus<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">h(x)=\\displaystyle\\frac{\\color{green}{x-1}}{\\color{red}{\\sqrt{x+1}}}<\/span>\n<p style=\"text-align: justify;\">Numerator annullatur in <span class=\"katex-eq\" data-katex-display=\"false\">x=1<\/span>; Denominator, praeterquam quod annullatur in <span class=\"katex-eq\" data-katex-display=\"false\">x=-1<\/span>, indeterminatus est si <span class=\"katex-eq\" data-katex-display=\"false\">x\\lt-1<\/span>. Ex his notitiis componitur sequens tabula signorum:<\/p>\n<table>\n<tbody>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\infty<\/span><\/th>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/th>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">+1<\/span><\/th>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">+\\infty<\/span><\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x-1<\/span><\/th>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\infty <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} +\\infty <\/span><\/td>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x+1}<\/span><\/th>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Non\\,Exsistit  <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Non\\,Exsistit <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} + <\/span><\/td>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{x-1}{\\sqrt{x+1}}<\/span><\/th>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Non\\,Exsistit <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{}Non\\,Exsistit <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> -\\infty <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} +\\infty <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ex informatione in hac tabula exposita, nunc valde simplex est graphium functionis delineare.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-mWc6Hza3Wl0\/YNWMYho7pPI\/AAAAAAAAFO4\/0D8zrIeKcc8HY7hlWuvJOWDnYE6Zw--cQCLcBGAsYHQ\/s0\/grafico%2B2.PNG\" alt=\"domainium, ambitus et graphium cum tabula signorum\" class=\" aligncenter lazyload\" width=\"498\" height=\"310\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-mWc6Hza3Wl0\/YNWMYho7pPI\/AAAAAAAAFO4\/0D8zrIeKcc8HY7hlWuvJOWDnYE6Zw--cQCLcBGAsYHQ\/s0\/grafico%2B2.PNG\" alt=\"domainium, ambitus et graphium cum tabula signorum\" class=\" aligncenter lazyload\" width=\"498\" height=\"310\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Et ita, determinare domainium et ambitum nunc est res trivialis:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Dom(h)=]-1,+\\infty[<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Rec(h)=\\mathbb{R}<\/span>\n<h3>Exercitium Propositum<\/h3>\n<p style=\"text-align: justify;\">Utens instrumentis quae modo revisimus, inveni domainium, ambitum et graphium sequentis functionis<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x) = \\displaystyle\\frac{4x^3 + 6x^2 -2x + 1}{x^2-4}<\/span>\n","protected":false},"excerpt":{"rendered":"<p>Domainium, Ambitus et Graphicum Functionum Algebraicarum Summarium: Haec lectio introducit notiones domainii, ambitus et graphicorum functionum, applicando eas ad exempla practica functionum algebraicarum. Examinantur technicae graphicae et analyticae ad hos elementa determinanda. Proposita Discendi: Post hanc lectionem discipulus poterit: Definire recte domainium, ambitum et graphium functionis. Adhibere methodos graphicas ad determinandum domainium et ambitum functionum [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29058,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33758","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Domainium, Ambitus et Graphicum Functionum Algebraicarum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Domainium, ambitus et graphicum functionis sunt notiones fundamentales in mathematica. 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