函数的周期和对称性

函数的周期和对称性 专题:函数的周期性对称性 1、周期函数的定义 一般地，对于函数，如果存在一个非零常数T，使得当x取定义域内的每一y,f(x) 个值时，都有，那么函数就叫做周期函数，非零常数T叫做这f(x，T),f(x)y,f(x) 个函数的一个周期。如果所有的周期中存在着一个最小的正数，就把这个最小的正数叫做最小正周期。 显然，若T是函数的周期，则也是的周期。如无特别说明，我们kT(k,z,k,0)f(x) 后面一般所说的周期是指函数的最小正周期。 说明:1、周期函数定义域必是无界的。 2、周期函数不一定都有最小正周期。 b,a推广:若，则是周期函数，是它的一个周期; f(x，a),f(x，b)f(x) TTf(x，),f(x,)，则周期为T; f(x)22 T的周期为的周期为。 fx()T,f(,x), 2、常见周期函数的函数方程: (1)函数值之和定值型，即函数 f(a，x)，f(b，x),C(a,b)对于定义域中任意x满足，则有，f(a，x)，f(b，x),C(a,b)f[x，(2b,2a)],f(x)故函数的周期是f(x)T,2(b,a) Ta,2fxafx，,,，，特例:，则fx是以为周期的周期函数; ，，，， (2)两个函数值之积定值型，即倒数或负倒数型 f(a，x),f(b，x),C(a,b，C可正可负)若，则得 f(x，2a),f[(x，2a)，(2b,2a)]f(x)T,2(b,a)，所以函数的周期是 in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the 1，f(x，b)(3)分式型，即函数满足 f(x，a),(a,b)f(x)1,f(x，b) ,11，f(x，b)由得，进而得 f(x，2a),f(x，a),(a,b)f(x，2b)1,f(x，b) ，由前面的结论得的周期是 f(x，2a),f(x，2b),,1f(x)T,4(b,a)特例: 1fxa，,,，则是以Ta,2为周期的周期函数; ，，fx，，fx，， 1T,3a，则是以为周期的周期函数. f(x，a),,，，fx1，f(x) 1T,3a，则是以为周期的周期函数. f(x，a),1,，，fxf(x) 1T,3a，则是以为周期的周期函数. f(x，a),，，fx1,f(x) 1()，fxTa,4，则是以为周期的周期函数. ，，fxfxa()，,1(),fx f(x),1Ta,4f(x，a),，则是以为周期的周期函数. ，，fxf(x)，1 f(x)，1T,2af(x，a),，则，，是以为周期的周期函数. fxf(x),1 1(),fxTa,2，，，则是以为周期的周期函数. fxfxa()，,1()，fx (4)递推型: (或)，则的周期T= 6a(联f(x，a),f(x),f(x,a)f(x),f(x,a),f(x,2a)f(x)系数列) fxfxafxafxafxa()()(2)(3)(4)，，，，，，，,，，，，fxfxafxafxafxa()()(2)(3)(4),则f(x) 的周期T=5a; ,12a其中，则y,f(x)是以为周g(x),g(x)y,f(x)满足f(x，a),g(f(x))，(a,0)， 期的周期函数。 in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the 3、函数的对称性与周期性之间的联系:双对称性函数的周期性 具有多重对称性的函数必具有周期性。即，如果一个函数有两条对称轴(或一条对称轴 和一个对称中心、或两个纵坐标相同的对称中心)，则该函数必为周期函数。 相关结论如下: xb,结论1:两线对称型:如果定义在上的函数有两条对称轴、，即Rxa,fx() ，且，那么是周期函数，其中一个周期fx()faxfax()()，,,fbxfbx()()，,, Tab,,2 证明:?得 faxfax()()，,,fxfax()(2),, 得 fbxfbx()()，,,fxfbx()(2),, ? faxfbx(2)(2),,, ? fxfbax()(22),,， 22ba,?函数是周期函数，且是一个周期。 yfx,() xb,【注意:上述2ab,不一定是最小正周期。若题目所给两条对称轴、之间没xa,有其他对称轴，则2ab,是最小正周期。具体可借助三角函数来进行分析。下同。】 ab,ac,bc,结论2:两点对称型:如果函数同时关于两点、()成中心对称，即，，，， 和，那么是周期函数，其中一faxfaxc()()2，，,,fbxfbxc()()2，，,,fx()()ab, Tab,,2个周期 证明:由 faxfaxc()()2，，,,fxfaxc()(2)2，,,, fbxfbxc()()2，，,,fxfbxc()(2)2，,, , 得faxfbx(2)(2),,, 得fxfbax()(22),,， 22ba,?函数yfx,()是以为周期的函数。 a,0ac,结论3:一线一点对称型:如果函数的图像关于点()成中心对称，且关fx()，， xb,ab,Tab,,4于直线()成轴对称，那么是周期函数，其中一个周期 fx() 证明: faxfaxcfxfaxc()()2()(2)2，，,,,，,, fbxfbxfxfbx()()()(2)，,,,,, fbaxfbabx(4())(2(42)),，,,,, fabxfabaxcfbax(42)(2(22))2(22),,,,,，,,,， ,,,,,,,2(2(2))2(2)cfbaxcfax ,,,,,，,2(2())22()()ccfxccfxfx a,0xa,推论1:如果偶函数fx()fx()的图像关于直线()对称，那么是周期函数， Ta,2其中一个周期 in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the3 a,0推论2:如果偶函数的图像关于直线()对称，那么是周期函数，ac,fx()fx()，， 其中一个周期 Ta,4 a,0推论3:如果奇函数的图像关于直线()对称，那么是周期函数，xa,fx()fx()其中一个周期 Ta,4 a,0推论4:如果奇函数关于点()成中心对称，那么是周期函数，其ac,fx()fx()，， 中一个周期Ta,2 【函数的奇偶性、对称性、周期性的代数特征有相仿之处，这三性都是有函数方程决定的，方程的不同特征决定了函数不同的性质，要注意其共性与个性。】 【函数的奇偶性是函数对称性中的特殊情况，奇函数对称中心为(0,0)，偶函数对称轴为y=0，带入结论1-3，可得推论1-4，所以学生在记忆时只需记住结论1-3即可，减少工作量】 【同理，教师可示范性给出一个结论的证明过程，其余可让学生进行证明】 典例精讲 一 利用周期性求值: 1例1、(??)函数对于任意实数满足条件，若，则f(x)f(x，2),f(1),,5xf(x) 1-=___ _____。 f(f(5))5 例2、(??)已知定义在R上的奇函数满足，则的值为 ( B f(x)f(x，2),,f(x)f(6) ) A、,1 B、0 C、1 D、2 例3、(??)已知奇函数满足f(x) xf(x，2),,f(x),且x,(0,1)时,f(x),2,则f(log18)的值为 。 1 2 解:，？fxfxfxfxfx(2)()(2)(4)，,,？,,，,，，， 99fffff(log18)(log18)(4log18)(log)(log),,,,,,, 12222882 9log9928,,,,,,f(log)2288 【提问:当所要求的值不在定义域中时，怎样通过变换将要求的函数值转化到已知解析式的这一段定义域中去,除了充分利用周期性外，还要注意题中的已知条件，如奇偶性、对称性等。】 in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the 例4、 (???)的定义域是，且,若 Rfx()fxfxfx(2)[1()]1()，,,，f(0)2008, 求 f(2008)的值。 fx(4)1，,,1fx(2)11，,,fx(4)1，，解:fxfx()(8),,,,，fx(4)1，, fxfx(2)1(4)，，，，1fx(4)1，， 周期为，8(2008)(0)2008？,,ff 二 利用周期性求解析式: 例5、(???)已知是以2为周期的偶函数，且当时，. fx()x,(0,1)fxx()1,， 求在上的解析式。 fx()(1,2) 解法1: 从解析式入手，由奇偶性结合周期性，将要求区间上问题转化为已知解析式的区间上 ? , 则 x,(1,2),,,,x(2,1) ?T,2, ? ，是偶函数 2(0,1),,x ? fxfxfxxx()()(2)213,,,,,,，,, x,(1,2) : 解法2 (从图象入手也可解决，且较直观) fxfx()(2),， 如图:, .?是偶函数 x,(0,1)fxx()1,， ?时 x,,(1,0)fxfxx()()1,,,,， 又周期为2，时 x,(1,2)x,,,2(1,0) ? fxfxxx()(2)(2)13,,,,,，,, T,5R例6、(???)已知函数是定义在上的周期函数，周期，函数yfx,() 是奇函数.又知在上是一次函数，在上是二次函yfx,()(11),,,xyfx,()[0,1][1,4] x,2,5数，且在时函数取得最小值. (1)证明:; ff(1)(4)0，, (2)求的解析式; yfxx,,(),[1,4] (3)求在上的解析式. yfx,()[4,9] 5解:?是以为周期的周期函数，且在上是奇函数，fx()[1,1], ?，?. ffff(1)(1)(51)(4),,,,,,,,ff(1)(4)0，, 2?当时，由题意可设， x,[1,4]fxaxa()(2)5 (0),,,, 22a,2由得，?， ff(1)(4)0，,aa(12)5(42)50,,，,,, 2?. fxxx()2(2)5(14),,,,, ??yfxx,,,,()(11)是奇函数，?f(0)0,， 又知yfx,()在[0,1]上是一次函数，?可设fxkxx()(01),,, 2而， f(1)2(12)53,,,,, k,,301,,x?，?当时，fxx()3,,， ,,,10x,,,11x从而fxfxx()()3,,,,,fxx()3,,时，，故时，. 46,,x,,,,151xfxfxxx()(5)3(5)315,,,,,,,，?当时，有，?. 69,,x154,,,x当时，， in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the5 22? fxfxxx()(5)2[(5)2]52(7)5,,,,,,,,, ,，,,315,46xx,?. fx(),,22(7)5,69xx,,,,, 【由以上两例可以看出，已知周期函数某个周期内的解析式，求另一个周期内的解析式，只要当成是函数图象的平移来做即可。】 【由于函数的性质:奇偶性、对称性联系紧密，教师可引导学生在此回顾奇函数、偶函数如何求解对称区间的解析式这类问题】 三 函数的奇偶性、对称性、周期性的综合运用 例1、(??)已知是定义在上的函数，且Rfx()fxfx(10)(10)，,, ，则( C ) fx()是fxfx(20)(20),,,， A. 周期为20的奇函数 B. 周期为20的偶函数 C. 周期为40的奇函数 D. 周期为40的偶函数 Rfxfxfx,,,，48yfx,，8例2、(??)定义域为的函数满足，且为偶，，，，，，，，函数，则( C ) fx() (A)是周期为4的周期函数 (B)是周期为8的周期函数 (C)是周期为12的周期函数 (D)不是周期函数 【熟记双对称函数的周期判断方法，如果学生对函数的对称性有所忘记，可在此回顾一下。】 R例3、(???)定义在上的函数，给出下列四个命题: fx() x,3(1)若是偶函数，则的图象关于直线对称 fx()fx(3)， (2)若则的图象关于点对称 fxfx(3)(3),，,,,fx()(3,0) (3)若=，且，则的一个周期为2。 fx(3)，fx(3),fxfx(4)(4)，,,fx() x,3(4)yfx,，(3)与yfx,,(3)的图象关于直线对称。 其中正确命题的序号为 (2)、(3) 。 【本题中学生容易错选(4)，究其原因，是没有分清是一个函数自身的对称，还是两个函数之间的对称，审题不清】 3,,fx,fx()fxfx(3)()0，，,(???)定义在上的函数满足，且函数为奇函R,,例4、2,,in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the 数(给出以下3个命题: ?函数的周期是6; fx() 3,,,，0?函数的图象关于点对称; fx(),,2,, ?函数的图象关于轴对称，其中，真命题的个数是( A )( fx()y A(3 B(2 C(1 D(0 例5、(???)设函数在上满足，，且fx()(,),,，,fxfx(2)(2),,，fxfx(7)(7),,，在闭区间,0，7,上，只有( ff(1)(3)0,, (?)试判断函数的奇偶性; yfx,() (?)试求方程=0在闭区间,-2005，2005,上的根的个数，并证明你的结论( fx() 解:由，得函数的对称轴为fxfx(2)(2),,，fxfx(7)(7),,，yfx,()xx,,27和, 从而知函数不是奇函数, yfx,() fxfxfxfx(2)(2)()(4),,，,,,,由 ,,,,,fxfx(4)(14),,fxfxfxfx(7)(7)()(14),,，,,,, T,10,从而知函数的周期为 ,,，fxfx()(10)yfx,() 又,故函数是非奇非偶函数; ffff(3)(1)0,(3)(7)0,,,,,而yfx,() fxfxfxfx(2)(2)()(4),,，,,,,(II)由 ,,,,,fxfx(4)(14),,fxfxfxfx(7)(7)()(14),,，,,,, ,,，fxfx()(10) III10[3,0]2[4,7]？函数周期为，关于对称的区间为，,,x，， ？fx()[0,7]13[4,7][3,0]在只有两根和，无根，无根，？？, ？,fx()[3,7]在一个周期内只有两根; ,？,2005,2005401()2005,2005802共有个周期，在共有个根。fx,,,, . 【本题的关键是要说明在一个周期内函数只有两个根，也就是函数在[7，10]内是无根的】 R例6 、(???)若函数在上是奇函数，且在上是增函数，且，，,1，0f(x) . f(x，2),,f(x) ?求的周期; f(x) xk,，21?证明的图象关于点中心对称;关于直线轴对称, ; f(x)(2,0)k()kZ, ?讨论在上的单调性; f(x)(1,2) T,4解: ?由已知，故周期. fxfxfxfx()(2)(22)(4),,，,，，,， P?设是图象上任意一点,则,且关于点对称的点为Pxy(,)yfx,()(2,0)k xk,，21Pkxy(4,),,Pkxy(42,)，,.P关于直线对称的点为 12 P?fkxfxfxy(4)()(),,,,,,,,?点在图象上,图象关于点(2,0)k对称. 1 又fx()是奇函数,fxfxfx(2)()()，,,,, ?fkxfxfxy(42)(2)()，,,,,, xk,，21P ?点在图象上,图象关于直线对称. 2 12,,,xx,,,,,,,21xx0221,,,,,xx?设，则， 122121 in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the7 ?在上递增, ?……(*) fxfx(2)(2),,,fx()(1,0),12 又 ?, . fxfx(2)(),,fxfx(2)(),,fxfxfx(2)()()，,,,,1122 所以: ，在上是减函数. fxfx()(),fx()(1,2)21 例7、(???)已知函数对任意实数均有xy,fx() xyxy，,，且存在非零常数 fxfyfff()()2()(),(0)0，,,cfc,()0.使,22 (1)求的值; f(0) (2)判断的奇偶性并证明; fx() (3)求证是周期函数，并求出的一个周期. fx()fx() 2解:(1)取，得abffffff,,，,,？,0,0(0)(0)2(0)(0),(0)(0)？ff(0)0,(0)1,？, (2)()fx是偶函数。 xxxx，,,,()()证明:原式中，不变，取得xyxfxfxff,,，,,,,()()2()()22即fxfxffxfxfxfx()()2(0)()2(),()()，,,,,？,, xx，(,,,2)(2)cxxc(3)令，则xxyxcfxfxcf=,()(2)2(,,，,,)()2()()0,,,,,ffxcfc22？，,,，？，，，,,？，，,fxfxcxxcfxcfxccfxcfx()(2)0,2(2)(22)0(2)()0将换成，， ？,,，，,，fxcfxcxxcfxfxc(2)(2),2()(4)再将换成，得 ？fxc()4的一个周期为 【本题第3问关键是如何利用这个条件，在等式右边凑出】 fc()0,fc()课堂检测 xx,0,11、(??)已知函数是以为周期的偶函数，且当时，，则 fx()fx()21,,2，，f(log10)的值为( ) 2 3835A.B.C.D., 5583 解:？3log104,14log100,,,？,,,,,22 ？函数是以为周期的偶函数，fx()2 ？,,，,,fff(log10)(4log10)(4log10), 222 13x？当是，xfxf,,,？,,，,,(0,1)()21,(4log10)161,2105 3即故选fA,(log10)..25 1xR,x,,,3,2fx(3)，,,fx()2、(??)设偶函数对任意，都有，且当时，,,fx()in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the ，则( D ) fxx()2,f(113.5), 2211A.B.C.D. ,,7755 xR,3、(??)设函数是定义在上的奇函数，对于任意的，都有fx()R 1(),fx0,x，当?时，，则( A ) fxx()2,f(11.5),fx(1)，,11()，fx 11A.B.C.D. , ,1122 01,,x4、(??)已知函数为上的奇函数，且满足，当时，f(x)fxfx(2)()，,,R ，则等于( B ) fxx(),f(7.5) A.0.5B.C.1.5D.,0.5,1.5 5、(??)设是定义在上的奇函数，且5，则Rfx()fxfx(4)()，,,f(3), ______________，______________ f(21),,f(2005), 答:5，,5 16、(??)设是定义在上的偶函数，且满足，当0?x?1，2x，Rfx()fx(2)，,fx(),fx()则______________ f(7.5), 答:1 7、(??)设是定义在上的奇函数，且， 2，则Rfx()fxfx(2)(2),,，f(1), ______________ ff(2)(7)，, 答:,2 fx,，1fx8、(???)设是定义域为R的函数，且fxfx，,21，又，，，，，，，，，，，， f2006，则= f222,，，，，， 22,(答:) 2 1，x9、(???)设f(x),，又记f(x),f(x)，f(x),f[f(x)]，k,1,2，…，则f(x),( D ) ，1k1k20091,x x,11，x1A(, B(x C. D. xx，11,x 【本题属于迭代周期型，也是常出现的题】 10、(???)已知定义在R上的奇函数f(x)，满足f(x,4),,f(x)，且在区间[0,2]上是增函 数，若方程f(x),m(m>0)在区间[,8,8]上有四个不同的根x，x，x，x，则x，x，x，x12341234 ,___-8_____. 11、(???)已知函数f(x)的定义域为R，且满足f(x，2),,f(x)( (1)求证:f(x)是周期函数; 11(2)若f(x)为奇函数，且当0?x?1时，f(x),x，求使f(x),,在[0,2009]上的所有x的个22 in southwest of Zhejiang Provincial Committee of the party's struggle, develop democratic United Front against Japan, established the armed forces and the anti-Japanese base area, decided to establish the CPC Zhejiang Committee. In February 1939, with Gu Yuliang, Secretary Zhu, as Minister, Jie Xu into a propaganda Minister, Zhang Zhihua as women's Minister, Huang yan as youth Minister, zhexi recorder of Penrhyn as military Secretary of the establishment. Hanchang organization attached to the recorders of the West. After the Spring Festival, Zhu Xi was determined to rally, led by Wang Hesong more than more than 100 people, with "first Road South into the team headquarters the name of" back from Wuzhen to Wujiang yan ... Is month, established the Tomb area 1th branch-East water Hong Kong Branch, Branch Secretary Shen Yingjie. Is autumn, the Kuomintang Zhejiang Provincial Government in Southwest Bureau to intensify its anti-communist, ordered political team training in tianmu mountain, is intended to eliminate in the political power of the Communist Party, thorough restructuring of the province's political team to make it become a tool of anti-Communist. In view of this situation, recorder of the West direction West open mass work has been difficult, political signs should not be used, only all go underground, the party's work to continue. To do this, take appropriate measures by a party organization, party members in the Organization of political withdrawal, transfer or make public career as cover, into a secret struggle. In November, Yang and the victory of the battle of the9 数( 解析:(1)证明 ?f(x，2),,f(x)， ?f(x，4),,f(x，2),,[,f(x)],f(x)， ?f(x)是以4为周期的周期函数( 1(2)当0?x?1时，f(x),x，设,1?x?0，则0?,x?1， 2 11?f(,x),(,x),,x. 22 ?f(x)是奇函数，?f(,x),,f(x)( 111?,f(x),,x，即f(x), x(,1?x?0)(故f(x), x(,1?x?1)( 222 1又设1,x,3，则,1,x,2,1，?f(x,2),(x,2)， 2又?f(x,2),,f(2,x),,f((,x)，2),,[,f(,x)] ,,f(x)， 11?,f(x),(x,2)，?f(x),,(x,2)(1,x,3)( 22 1x，,,1?x?1,,21? f(x),. 由f(x),,，解得x,,1. ,21 ,x,2,，,1