新高考數(shù)學一輪復習第2章 第03講 函數(shù)的奇偶性 對稱性與周期性 精講+精練（學生版）

第03講函數(shù)的奇偶性、對稱性與周期性(精講+精練）目錄第一部分：知識點精準記憶第二部分：課前自我評估測試第三部分：典型例題剖析高頻考點一：函數(shù)奇偶性①判斷函數(shù)奇偶性②根據(jù)函數(shù)奇偶性求解析式③函數(shù)奇偶性的應用④由函數(shù)奇偶性求參數(shù)⑤奇偶性+單調性解不等式高頻考點二：函數(shù)周期性及其應用①由函數(shù)周期性求函數(shù)值②由函數(shù)周期性求解析式高頻考點三：函數(shù)的對稱性①由函數(shù)對稱性求解析式②由函數(shù)對稱性求函數(shù)值或參數(shù)③對稱性+奇偶性+周期性的綜合應用第四部分：高考真題感悟第五部分：第03講函數(shù)的奇偶性、對稱性與周期性（精練）第一部分：知識點精準記憶第一部分：知識點精準記憶1、函數(shù)的奇偶性（1）函數(shù)奇偶性定義奇偶性定義圖象特點偶函數(shù)如果對于函數(shù)SKIPIF1<0的定義域內任意一個SKIPIF1<0，都有SKIPIF1<0，那么函數(shù)SKIPIF1<0是偶函數(shù)圖象關于SKIPIF1<0軸對稱奇函數(shù)如果對于函數(shù)SKIPIF1<0的定義域內任意一個SKIPIF1<0，都有SKIPIF1<0，那么函數(shù)SKIPIF1<0是奇函數(shù)圖象關于原點對稱注意：由函數(shù)奇偶性的定義可知，函數(shù)具有奇偶性的一個前提條件是：對于定義域內的任意一個x，SKIPIF1<0也在定義域內（即定義域關于原點對稱）．（2）常用結論與技巧：①對數(shù)型復合函數(shù)判斷奇偶性常用SKIPIF1<0或SKIPIF1<0來判斷奇偶性.②SKIPIF1<0，SKIPIF1<0在它們的公共定義域上有下面的結論：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)奇函數(shù)不能確定不能確定奇函數(shù)奇函數(shù)奇函數(shù)偶函數(shù)不能確定不能確定奇函數(shù)奇函數(shù)奇函數(shù)奇函數(shù)奇函數(shù)奇函數(shù)偶函數(shù)偶函數(shù)③若SKIPIF1<0是定義在區(qū)間SKIPIF1<0上奇函數(shù)，且SKIPIF1<0，則SKIPIF1<0（注意：反之不成立）2、函數(shù)對稱性（異號對稱）（1）軸對稱：若函數(shù)SKIPIF1<0關于直線SKIPIF1<0對稱，則①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0（2）點對稱：若函數(shù)SKIPIF1<0關于直線SKIPIF1<0對稱，則①SKIPIF1<0②SKIPIF1<0③SKIPIF1<0（2）點對稱：若函數(shù)SKIPIF1<0關于直線SKIPIF1<0對稱，則①SKIPIF1<0②SKIPIF1<0③SKIPIF1<03、函數(shù)周期性（同號周期）（1）周期函數(shù)定義對于函數(shù)SKIPIF1<0，如果存在一個非零常數(shù)SKIPIF1<0，使得當SKIPIF1<0取定義域內的任何值時，都有SKIPIF1<0，那么就稱函數(shù)SKIPIF1<0為周期函數(shù)，稱SKIPIF1<0為這個函數(shù)的周期，則SKIPIF1<0（SKIPIF1<0）也是這個函數(shù)的周期.（2）最小正周期如果在周期函數(shù)SKIPIF1<0的所有周期中存在一個最小的正數(shù)，那么這個最小的正數(shù)就叫做SKIPIF1<0的最小正周期（若不特別說明，SKIPIF1<0一般都是指最小正周期）.注意：并不是所有周期函數(shù)都有最小正周期.（3）函數(shù)周期性的常用結論與技巧設函數(shù)SKIPIF1<0，SKIPIF1<0.①若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；②若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；③若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；④若SKIPIF1<0，則函數(shù)的周期SKIPIF1<0；⑤SKIPIF1<0，則函數(shù)的周期SKIPIF1<0第二部分：課前自我評估測試第二部分：課前自我評估測試1．（2022·北京·高三學業(yè)考試）已知函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0是奇函數(shù) B．SKIPIF1<0是偶函數(shù)C．SKIPIF1<0既是奇函數(shù)又是偶函數(shù) D．SKIPIF1<0既不是奇函數(shù)也不是偶函數(shù)2．（2022·浙江臺州·高一期末）設f（x）是定義在R上的奇函數(shù)，若SKIPIF1<0，則f（1）=（

）A．-1 B．0 C．1 D．23．（2022·全國·高三專題練習）若SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且SKIPIF1<0，則SKIPIF1<0的值為（

）A．1 B．2 C．0 D．SKIPIF1<04．（2021·全國·高一課時練習）若SKIPIF1<0的偶函數(shù)，其定義域為SKIPIF1<0，且在SKIPIF1<0上是減函數(shù)，則SKIPIF1<0與SKIPIF1<0得大小關系是A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．不能確定5．（2021·河南·新蔡縣第一高級中學高三階段練習（文））已知函數(shù)f(x)為定義在R上的奇函數(shù)，且SKIPIF1<0，則SKIPIF1<0（

）A．2019 B．3 C．－3 D．0第三部分：典型例題剖析第三部分：典型例題剖析高頻考點一：函數(shù)奇偶性①判斷函數(shù)奇偶性1．（2021·廣東·汕頭市潮陽區(qū)河溪中學高二期中）下列函數(shù)在其定義域內為奇函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2021·江蘇·高一單元測試）函數(shù)SKIPIF1<0為奇函數(shù)，SKIPIF1<0為偶函數(shù)，在公共定義域內，下列結論一定正確的是（

）A．SKIPIF1<0為奇函數(shù) B．SKIPIF1<0為偶函數(shù)C．SKIPIF1<0為奇函數(shù) D．SKIPIF1<0為偶函數(shù)3．（2021·廣東·龍門縣高級中學高一期中）給定函數(shù)：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0.其中奇函數(shù)是（

）.A．①② B．③④ C．②④ D．①③②根據(jù)函數(shù)奇偶性求解析式1．（2021·四川省南充高級中學高一階段練習）若函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，則該函數(shù)的最大值為（

）A．10 B．5 C．3 D．22．（2021·寧夏·銀川一中高一期中）已知SKIPIF1<0是定義域為R的偶函數(shù)，當SKIPIF1<0時，SKIPIF1<0，則函數(shù)SKIPIF1<0在SKIPIF1<0時，SKIPIF1<0=___________.3．（2021·江蘇·南京外國語學校高一期中）設m為實數(shù)，若函數(shù)SKIPIF1<0（SKIPIF1<0）是偶函數(shù)，則m的值為__________.4．（2021·全國·高一課前預習）已知SKIPIF1<0是SKIPIF1<0上的奇函數(shù)，且當SKIPIF1<0時，SKIPIF1<0，求SKIPIF1<0的解析式．③函數(shù)奇偶性的應用1．（2022·湖南·長沙市南雅中學高三階段練習）設f(x)是周期為2的奇函數(shù)，當0≤x≤1時，f(x)＝2x，則SKIPIF1<0＝________.2．（2022·廣東茂名·高一期末）若函數(shù)SKIPIF1<0是奇函數(shù)，則SKIPIF1<0__________.3．（2022·四川涼山·高一期末）已知SKIPIF1<0，SKIPIF1<0分別是定義在R上的偶函數(shù)和奇函數(shù)，且SKIPIF1<0，則SKIPIF1<0______.4．（2022·湖南·一模）已知SKIPIF1<0是奇函數(shù)，且SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0___．④由函數(shù)奇偶性求參數(shù)1．（2022·內蒙古包頭·高三期末（文））已知函數(shù)SKIPIF1<0是偶函數(shù)，則SKIPIF1<0______．2．（2022·海南·模擬預測）已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，則SKIPIF1<0______.3．（2022·湖北·石首市第一中學高一階段練習）已知函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0_______．4．（2022·黑龍江·佳木斯一中高一期末）SKIPIF1<0為偶函數(shù)，則SKIPIF1<0___________.⑤奇偶性+單調性解不等式1．（2022·廣西南寧·高一期末）若函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，在SKIPIF1<0上單調遞減，且SKIPIF1<0，則使得SKIPIF1<0的SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0D．SKIPIF1<02．（2022·云南麗江·高一期末）已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．（2022·四川綿陽·高一期末）若SKIPIF1<0，則滿足SKIPIF1<0的SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．（2022·廣東汕尾·高一期末）函數(shù)SKIPIF1<0為奇函數(shù)，且對任意互不相等的SKIPIF1<0，SKIPIF1<0，都有SKIPIF1<0成立，且SKIPIF1<0，則SKIPIF1<0的解集為______．5．（2022·甘肅省武威第一中學高一開學考試）設偶函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調遞增，則滿足SKIPIF1<0的x的取值范圍是___________.6．（2022·湖北大學附屬中學高一階段練習）SKIPIF1<0是奇函數(shù)(1)求SKIPIF1<0(2)判斷并證明SKIPIF1<0的單調性(3)若SKIPIF1<0，求SKIPIF1<0的取值范圍高頻考點二：函數(shù)周期性及其應用①由函數(shù)周期性求函數(shù)值1．（2021·北京·人大附中高一期中）已知定義在SKIPIF1<0上的奇函數(shù)，SKIPIF1<0滿足SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·甘肅·一模（文））定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0，滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．8 B．2 C．-2 D．-83．（2021·廣東汕頭·高二期末）已知函數(shù)SKIPIF1<0是奇函數(shù)，且滿足SKIPIF1<0，若當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0________.②由函數(shù)周期性求解析式1．（2021·北京市十一學校高一期中）若定義在R上的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0時SKIPIF1<0，則：（1）SKIPIF1<0__________；（2）當SKIPIF1<0時，SKIPIF1<0_________.2．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0是定義域為R的偶函數(shù)，且周期為2，當SKIPIF1<0時，SKIPIF1<0，則當SKIPIF1<0時，SKIPIF1<0________.3．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0對任意實數(shù)SKIPIF1<0都有SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0.（1）求SKIPIF1<0，SKIPIF1<0的值；（2）寫出SKIPIF1<0在SKIPIF1<0，SKIPIF1<0上的解析式；（3）當SKIPIF1<0，SKIPIF1<0時，求不等式SKIPIF1<0的解集.4．（2021·山東師范大學附中高三期中）設SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且對任意實數(shù)SKIPIF1<0，恒有SKIPIF1<0.當SKIPIF1<0時，SKIPIF1<0.(1)當SKIPIF1<0時，求SKIPIF1<0的解析式；(2)計算SKIPIF1<0.高頻考點三：函數(shù)的對稱性①由函數(shù)對稱性求解析式1．（2022·廣東·高三開學考試）下列函數(shù)與SKIPIF1<0關于SKIPIF1<0對稱的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·浙江·高三專題練習）已知函數(shù)SKIPIF1<0的圖象與函數(shù)SKIPIF1<0的圖象關于SKIPIF1<0軸對稱，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0滿足：①SKIPIF1<0；②在SKIPIF1<0上是減函數(shù)；③SKIPIF1<0.請寫出一個滿足以上條件的SKIPIF1<0___________.4．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0，滿足SKIPIF1<0，則SKIPIF1<0______.③由函數(shù)對稱性求函數(shù)值或參數(shù)1．（2021·江西·景德鎮(zhèn)一中高二期末（文））已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．2 B．SKIPIF1<0 C．4 D．SKIPIF1<02．（2021·全國·高一專題練習）已知函數(shù)SKIPIF1<0，記SKIPIF1<0SKIPIF1<0，則SKIPIF1<0A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·四川雅安·高一期末）若SKIPIF1<0，則SKIPIF1<0___________.4．（2021·上?！じ咭粚ｎ}練習）SKIPIF1<0的對稱中心為SKIPIF1<0，則a的值為___________．5．（2021·全國·高一專題練習）已知函數(shù)f（x）＝SKIPIF1<0.（1）求f（2）與fSKIPIF1<0，f（3）與fSKIPIF1<0；（2）由（1）中求得的結果，你能發(fā)現(xiàn)f（x）與fSKIPIF1<0有什么關系？證明你的發(fā)現(xiàn)；（3）求f（2）＋fSKIPIF1<0＋f（3）＋fSKIPIF1<0＋SKIPIF1<0＋f（2019）＋fSKIPIF1<0的值.④對稱性+奇偶性+周期性的綜合應用1．（2022·四川涼山·二模（文））定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0，滿足SKIPIF1<0，當SKIPIF1<0時SKIPIF1<0，則SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·重慶·西南大學附中模擬預測）函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則關于x的方程SKIPIF1<0在SKIPIF1<0上的解的個數(shù)是（

）A．1010 B．1011 C．1012 D．10133．(多選）（2022·甘肅·蘭州一中高一期末）定義在R上的偶函數(shù)f（x）滿足SKIPIF1<0，且在SKIPIF1<0上是增函數(shù)，則下列關于f（x）的結論中正確的有（

）A．f（x）的圖象關于直線SKIPIF1<0對稱 B．f（x）在[0，1]上是增函數(shù)C．f（x）在[1，2]上是減函數(shù) D．SKIPIF1<04．(多選）（2022·全國·高三專題練習）已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則（

）A．SKIPIF1<0為奇函數(shù) B．SKIPIF1<0的圖象關于SKIPIF1<0對稱C．SKIPIF1<0為偶函數(shù) D．SKIPIF1<0是周期為4的函數(shù)5．（2022·重慶九龍坡·高一期末）若函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0時，SKIPIF1<0，已知函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內的零點的個數(shù)為__________.第四部分：高考真題感悟第四部分：高考真題感悟1．（2021·全國·高考真題（文））設SKIPIF1<0是定義域為R的奇函數(shù)，且SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2021·全國·高考真題）已知函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，SKIPIF1<0為偶函數(shù)，SKIPIF1<0為奇函數(shù)，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2021·江蘇·高考真題）已知奇函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的單調函數(shù)，若正實數(shù)SKIPIF1<0，SKIPIF1<0滿足SKIPIF1<0則SKIPIF1<0的最小值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．44．（2021·全國·高考真題（理））設函數(shù)SKIPIF1<0的定義域為R，SKIPIF1<0為奇函數(shù)，SKIPIF1<0為偶函數(shù)，當SKIPIF1<0時，SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2021·湖南·高考真題）已知函數(shù)SKIPIF1<0為奇函數(shù)，SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0____________6．（2021·全國·高考真題）已知函數(shù)SKIPIF1<0是偶函數(shù)，則SKIPIF1<0______.第五部分：第03講函數(shù)的奇偶性、對稱性與周期性（精練）第五部分：第03講函數(shù)的奇偶性、對稱性與周期性（精練）1．（2022·山西·懷仁市第一中學校二模（理））已知函數(shù)SKIPIF1<0為R上的奇函數(shù)，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0等于（

）A．-3 B．-1 C．1 D．32．（2022·山西呂梁·一模（文））已知函數(shù)SKIPIF1<0為定義在R上的奇函數(shù)，且當SKIPIF1<0時，SKIPIF1<0，則當SKIPIF1<0時，SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．（2022·江蘇·南京師大附中高一期末）定義在SKIPIF1<0上的偶函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調遞增，若SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022·全國·高三專題練習（文））已知定義在SKIPIF1<0上的偶函數(shù)SKIPIF1<0，對SKIPIF1<0，有SKIPIF1<0成立，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0的圖象關于原點對稱，且滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022·陜西咸陽·二模（理））已知函數(shù)SKIPIF1<0為定義在R上的奇函數(shù)，且SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．2021 B．1 C．SKIPIF1<0 D．07．（2022·山西·懷仁市第一中學校二模（文））已知SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，SKIPIF1<0為偶函數(shù)，且當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．4 B．3 C．2 D．18．（2022·廣東·執(zhí)信中學高一階段練習）已知在R上的函數(shù)SKIPIF1<0滿足對于任意實數(shù)SKIPIF1<0都有SKIPIF1<0，SKIPIF1<0，且在區(qū)間SKIPIF1<0上只有SKIPIF1<0和SKIPIF1<0兩個零點，則SKIPIF1<0在區(qū)間SKIPIF1<0上根的個數(shù)為（）A．404 B．405 C．406 D．203二、填空題9．（2022·上海市復興高級中學高一階段練習）已知SKIPIF1<0，若SKIPIF1<0，則實數(shù)SKIPIF1<0的取值范圍是______10．（2022·江西·新余市第一中學高一開學考試）已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0________．11．（2022·重慶巴蜀中學高一期末）已知定義在區(qū)間