奇函数和偶函数讲稿

奇函数和偶函数讲稿 函数的奇偶性讲稿 (一、导入新课) 现在开始上课，今天我为大家讲解一下有关函数奇偶性的概念以及如何判断函数奇偶性。 在此之前，先回忆一下之前讲的有关对称的概念，我们会发现生活中有很多对称的例子。例如:汽车车轮，人(一般只要是圆柱，圆锥，球，正方体，长方体几何体都是轴对称图形)，篮球，羽毛球拍等. 而数学中也存在对称的例子，例如今天所要讲的奇函数和偶函数。大家可以在纸上画出函数y=x，y=1/x，y=cos x ,y=x?的图象，看一下这些函数有什么特点。 (y=x，y=1/x图象关于原点对称,=cos x ,y=x?的图象关于y轴对称)。 (二、讲解新课) 如何从数值角度研究对称函数图象的自变量与函数值之间的规律。 下面以函数y=x?为例(画出函数图象)，首先我们知道,对于任意x,-x与x关于y轴对称，即x?与(,x)?两点到坐标y轴的距离相等，而且x?=(-x)?，也就是说函数y=x?的定义域上每一点都成立x?=(,x)?，而这样的函数我们通常称之为偶函数。 所以可以给出偶函数的定义:一般地,如果对于函数f(x)的定义域内任意一个x,都有f(x)=f(,x),那么函数f(x)就叫做偶函数. 注意“任意”两字。 (让大家举出一些偶函数的例子)既然关于y轴对称的函数我们称为偶函数，那么关于原点对称的函数呢,当然也有一个特定称谓叫做奇函数。而奇函数的自by Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, serious party political life, and strict party of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thought political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine theory confidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively soft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, normal life together, some party members and cadres is not required to participate in the party's organisational life. Grass-roots party members and cadres are more depressed, avoid responsibility, self-discipline is not strict, and not as, slow as. These issues relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Central situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occasions "two" study and education made important instructions, profoundly illustrates the significance of education. XI General Secretary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General members expand, from concentrated sex education to regular education extends, firm General members of Marx doctrine 变量与函数值之间具有怎样的数值规律呢?可以以函数y=1/x为例(同时画出出y=1/x的图象)， 我们可以类似的方法,得出函数y=1/x的定义域上每一点都成立1/x=-1/(-x),所以奇函数的定义. 一般地,如果对于函数f(x)的定义域内任意一个x,都有f(,x)= ,f(x),那么函数f(x)就叫做奇函数. 下面如何判定函数奇偶性, (三、例题讲解 写下:例1 判断下列函数的奇偶性 (1)f(x)=x，1/x; (2) f(x)= 1,x ?; (3) f(x)=2x ; (4) f(x)= |x|,2; 21/2(5)f(x)=(1,x); (6)f(x)= ,x ?,,3?x?1; (7) f(x)=2x,1;) 前三个题做完，可以发现判断奇偶性,只需验证 f(x)与f(,x)之间的关系.那如何判断一个函数不具有奇偶性呢,以第(1)为例,说说它为什么不是偶函数呢?(因为f(x)?f(,x))，所以判断一个函数不具有奇偶性只需举一个反例就可说明. 另一个需要注意的是，通过第(6)题我们可以得出:定义域关于原点对称是函数具有奇偶性的先决条件。 在这几个函数中有是奇函数不是偶函数,有是偶函数不是奇函数也有既不是奇函数也不是偶函数,那么有没有这样的函数,它既是奇函数也是偶函数呢? by Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, serious party political life, and strict party of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thought political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine theory confidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively soft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, normal life together, some party members and cadres is not required to participate in the party's organisational life. Grass-roots party members and cadres are more depressed, avoid responsibility, self-discipline is not strict, and not as, slow as. These issues relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Central situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occasions "two" study and education made important instructions, profoundly illustrates the significance of education. XI General Secretary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General members expand, from concentrated sex education to regular education extends, firm General members of Marx doctrine 2009220146 (当然有，例如函数f(x)=0)。那是不是具备这样性质的函数的解析式都只能写成这样呢?我们可以用下面这个例题来。 (例2 已知函数f(x)既是奇函数也是偶函数,求证: f(x)=0. 证明:?f(x)既是奇函数也是偶函数, ?f(,x)= f(x),且f(,x)= ,f(x) ?f(x)= ,f(x) 即2f(x)= 0;?f(x)=0) 我们可以再想一想:这样的函数应有多少个呢? (学生开始可能认为只有一个,经提示可发现, f(x)=0是解析式的特征,若改变函数的定义域,如f(x)=0,x?[,1,1],f(x)=0，x?,,2, ,1,0,1,2,,它们 显然是不同的函数,但它们都是既是奇函数也是偶函数.) 今天这一节我们主要介绍了函数奇偶性的定义及判定，而且知道利用函数的奇偶性还可将函数分为奇函数、偶函数、非奇非偶函数、既奇又偶函数.同学们还有什么问题, 那么这节课就先讲到这里，今天的作业是P1、2题; P 6题. 36 37(下课) 函数的奇偶性教案 课题类型 新知课 教学方法 讲解法、数形结合法 ions "expand, from concentrated sex education to regular education extends, firm General members of Marx doctrine embersary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General mtwo" study and education made important instructions, profoundly illustrates the significance of education. XI General Secretl situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occass relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Centradiscipline is not strict, and not as, slow as. These issue-roots party members and cadres are more depressed, avoid responsibility, self-life together, some party members and cadres is not required to participate in the party's organisational life. Grass normalsoft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, onfidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively eory cht political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine thty of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thougrious party political life, and strict parby Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, se3 教学目标 从形和数两个方面进行引导，使学生理解函数奇偶性的概念; 会利用定义判断简单函数的奇偶性. 教学重难点 教学重点: 函数奇偶性概念的形成与函数奇偶性的判断. 教学难点: 对函数奇偶性的概念的理解 教具 板书 教学过程 (一)导入新课 先举现实生活中对称美的例子,然后告诉学生数学中也存在这种对称美,试让学生举例. (学生可能会举出y=x和y=1/x,y=,x等例子)其中哪些函数的图象关于y轴对称, 以函数y=x?为例，画出图象,让学生说出判断其图象关于y轴对称的方法. 在数学上将图象关于y轴对称的函数叫做偶函数.今天将从数值角度研究图象关于y轴对称函数的自变量与函数值之间的规律. (二)讲解新课 引导学生先将规律具体化,再用数学符号表示.从而发现对定义域内任意一个x,都有f(x)=f(,x)成立.最后让学生用完整的语言给出偶函数定义,不准确的地方予以提示或调整. 一般地,如果对于函数f(x)的定义域内任意一个x,都有f(x)=f(,x),那么函数f(x)就叫做偶函数. by Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, serious party political life, and strict party of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thought political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine theory confidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively soft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, normal life together, some party members and cadres is not required to participate in the party's organisational life. Grass-roots party members and cadres are more depressed, avoid responsibility, self-discipline is not strict, and not as, slow as. These issues relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Central situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occasions "two" study and education made important instructions, profoundly illustrates the significance of education. XI General Secretary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General members expand, from concentrated sex education to regular education extends, firm General members of Marx doctrine 2009220146 注:强调“任意”两字. 给出定义后可让学生举例检验他们对概念的初步认识 提出新问题:图象关于原点对称的函数的自变量与函数值之间具有怎样的数值规律呢?(同时画出y=1/x的图象让学生观察研究) 引导学生用类比的方法,得出结论,让学生给出奇函数的定义. 一般地,如果对于函数f(x)的定义域内任意一个x,都有f(,x)= ,f(x),那么函数f(x)就叫做奇函数. (三) 例题讲解 例1 判断下列函数的奇偶性 (1)f(x)=x，1/x; (2) f(x)= 1,x ?; (3) f(x)=2x ; (4) f(x)= |x|,2; 21/2(5)f(x)=(1,x); (6)f(x)= ,x ?,,3?x?1; (7) f(x)=2x,1; 前三个题做完，进行一次小结,判断奇偶性,只需验证 f(x)与f(,x)之间的关系.此时提出问题如何判断一个函数不具有奇偶性呢,以第(1)为例,说明怎样解决它不是偶函数的问题呢?引导学生得出只需举一个反例就可说明. 通过第(6)题引导学生得出定义域关于原点对称是函数具有奇偶性的先决条件的结论. 由学生小结判断奇偶性的步骤之后,提出新的问题:在刚才的几个函数中有是奇函数不是偶函数,有是偶函数不是奇函数,也有既不是奇函数也不是偶函数,那么有没有这样的函数,它既是奇函数也是偶函数呢?若有,举例说明. ary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General mexpand, from concentrated sex education to regular education extends, firm General members of Marx doctrinetwo" study and education made important instructions, profoundly illustrates the significance of education. XI General Secret embersions "l situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occass relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Centradiscipline is not strict, and not as, slow as. These issue-roots party members and cadres are more depressed, avoid responsibility, self-life together, some party members and cadres is not required to participate in the party's organisational life. Grass normalsoft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, onfidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively eory cht political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine thty of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thougrious party political life, and strict parby Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, se5 经学生思考,可找到函数f(x)=0.然后继续提问:是不是具备这样性质的函数 的解析式都只能写成这样呢?能证明吗? 例2 已知函数f(x)既是奇函数也是偶函数,求证: f(x)=0. 证明:?f(x)既是奇函数也是偶函数, ?f(,x)= f(x),且f(,x)= ,f(x) ?f(x)= ,f(x) 即2f(x)= 0;?f(x)=0 进一步提问:这样的函数应有多少个呢? (学生开始可能认为只有一个,经提示可发现, f(x)=0是解析式的特征,若改变函数的定义域,如f(x)=0,x?[,1,1],f(x)=0，x?,,2, ,1,0,1,2,,它们 显然是不同的函数,但它们都是既是奇函数也是偶函数.) 小结 函数奇偶性的定义; 函数奇偶性的判定; 利用函数的奇偶性可将函数分为四类:奇函数、偶函数、非奇非偶函数、既奇又偶函数. 作业 P1、2题; P 6题. 36 37 by Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, serious party political life, and strict party of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thought political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine theory confidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively soft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, normal life together, some party members and cadres is not required to participate in the party's organisational life. Grass-roots party members and cadres are more depressed, avoid responsibility, self-discipline is not strict, and not as, slow as. These issues relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Central situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occasions "two" study and education made important instructions, profoundly illustrates the significance of education. XI General Secretary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General members expand, from concentrated sex education to regular education extends, firm General members of Marx doctrine 2009220146 函数的奇偶性 例1 判断下列函数的奇 注:定义域关于原点对称是函数 偶性 具有奇偶性的先决条件。 (1)f(x)=x，1/x; 4、存在 既不是奇函数也不是偶函 (2) f(x)= 1,x ?; 数? y=x? (3)f(x)=2x ; f(x)=0 1、一般地,如果对于函数 (4) f(x)= |x|,2; 例2 已知函数f(x)既是奇函数f(x)的定义域内任意一个x,21/2(5)f(x)=(1,x); 也是偶函数,求证: f(x)=0. 都有f(x)=f(,x),那么函数 (6)f(x)= ,x ?,,3?x 证明:?f(x)既是奇函数也 f(x)就叫做偶函数. ?1; 是偶函数, (7) f(x)=2x,1; ?f(,x)= f(x),且f(,x)= 解:(1)奇函数 ,f(x) y=1/x (2)偶函数 ?f(x)= ,f(x) 2、一般地,如果对于函数 (3) 奇函数 ; 即2f(x)= 0;?f(x)=0 f(x)的定义域内任意一个x, (4) 偶函数; 都有f(x)=-f(,x),那么函数 (5) 偶函数; f(x)=0,x?[,1,1]; f(x)就叫做奇函数. f(x)=0，x?,,2, ,1,0,1,2, (6)既不是奇函数也不是 偶函数; 3、函数奇偶性的判定 (7) 既不是奇函数也不 f(,x)=f(x) f(x)是, 是偶函数. P1、2题; P 6题. 36 37偶函数; f(,x)=,f(x) f(x), 是奇函数 ary stressed, deployment "two learn a do" learning education, is to promoted party education from "key minority" to General mexpand, from concentrated sex education to regular education extends, firm General members of Marx doctrinetwo" study and education made important instructions, profoundly illustrates the significance of education. XI General Secret embersions "l situation, decided to all party members in the "two" study and education. This year, the XI General Secretary on many occass relate to ignore party rules the party Constitution study, lack of knowledge of the party have a great relationship. Centradiscipline is not strict, and not as, slow as. These issue-roots party members and cadres are more depressed, avoid responsibility, self-life together, some party members and cadres is not required to participate in the party's organisational life. Grass normalsoft, "three lessons" implementation of systems are not in place, the party's political life is not serious and not serious, onfidence insufficient, on China features Socialist faith not firm. Some grassroots party organizations are still relatively eory cht political Shang problem also compared highlight, some belief confused, on Communist belief not really, on Marx doctrine thty of discipline and rules, and strengthening members cadres education and management, task also is heavy, some members thougrious party political life, and strict parby Western bad ideology of effect, community value orientation increasingly diversified, social thought divergent surging, se7