新高考數(shù)學(xué)一輪復(fù)習(xí)講義 第08講 函數(shù)的基本性質(zhì)Ⅱ-奇偶性、周期性和對稱性（原卷版）

第08講函數(shù)的基本性質(zhì)Ⅱ-奇偶性、周期性和對稱性（精講）題型目錄一覽①函數(shù)的奇偶性②函數(shù)奇偶性的應(yīng)用③函數(shù)的周期性④函數(shù)的對稱性⑤函數(shù)性質(zhì)的綜合應(yīng)用一、知識點梳理一、知識點梳理1.函數(shù)的奇偶性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(nèi)任意一個SKIPIF1<0，都有SKIPIF1<0，那么函數(shù)SKIPIF1<0就叫做偶函數(shù)關(guān)于SKIPIF1<0軸對稱奇函數(shù)如果對于函數(shù)SKIPIF1<0的定義域內(nèi)任意一個SKIPIF1<0，都有SKIPIF1<0，那么函數(shù)SKIPIF1<0就叫做奇函數(shù)關(guān)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(nèi)的任意一個SKIPIF1<0，SKIPIF1<0也在定義域內(nèi)(即定義域關(guān)于原點對稱)．2.函數(shù)的對稱性(1)若函數(shù)SKIPIF1<0為偶函數(shù)，則函數(shù)SKIPIF1<0關(guān)于SKIPIF1<0對稱．(2)若函數(shù)SKIPIF1<0為奇函數(shù)，則函數(shù)SKIPIF1<0關(guān)于點SKIPIF1<0對稱．(3)若SKIPIF1<0，則函數(shù)SKIPIF1<0關(guān)于SKIPIF1<0對稱．(4)若SKIPIF1<0，則函數(shù)SKIPIF1<0關(guān)于點SKIPIF1<0對稱．3.函數(shù)的周期性（1）周期函數(shù)：對于函數(shù)SKIPIF1<0，如果存在一個非零常數(shù)SKIPIF1<0，使得當(dāng)SKIPIF1<0取定義域內(nèi)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.（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)，那么稱這個最小整數(shù)叫做SKIPIF1<0的最小正周期.【常用結(jié)論】1.奇偶性技巧（1）若奇函數(shù)SKIPIF1<0在SKIPIF1<0處有意義，則有SKIPIF1<0；（2）對于運(yùn)算函數(shù)有如下結(jié)論：奇SKIPIF1<0奇=奇；偶SKIPIF1<0偶=偶；奇SKIPIF1<0偶=非奇非偶；奇SKIPIF1<0奇=偶；奇SKIPIF1<0偶=奇；偶SKIPIF1<0偶=偶.(3)常見奇偶性函數(shù)模型奇函數(shù)：=1\*GB3①函數(shù)SKIPIF1<0或函數(shù)SKIPIF1<0．=2\*GB3②函數(shù)SKIPIF1<0．=3\*GB3③函數(shù)SKIPIF1<0或函數(shù)SKIPIF1<0=4\*GB3④函數(shù)SKIPIF1<0或函數(shù)SKIPIF1<0．注意：關(guān)于=1\*GB3①式，可以寫成函數(shù)SKIPIF1<0或函數(shù)SKIPIF1<0．偶函數(shù)：=1\*GB3①函數(shù)SKIPIF1<0．=2\*GB3②函數(shù)SKIPIF1<0．=3\*GB3③函數(shù)SKIPIF1<0類型的一切函數(shù)．2.周期性技巧SKIPIF1<03.函數(shù)的的對稱性與周期性的關(guān)系（1）若函數(shù)SKIPIF1<0有兩條對稱軸SKIPIF1<0，SKIPIF1<0，則函數(shù)SKIPIF1<0是周期函數(shù)，且SKIPIF1<0；（2）若函數(shù)SKIPIF1<0的圖象有兩個對稱中心SKIPIF1<0，則函數(shù)SKIPIF1<0是周期函數(shù)，且SKIPIF1<0；（3）若函數(shù)SKIPIF1<0有一條對稱軸SKIPIF1<0和一個對稱中心SKIPIF1<0，則函數(shù)SKIPIF1<0是周期函數(shù)，且SKIPIF1<0.4.對稱性技巧（1）若函數(shù)SKIPIF1<0關(guān)于直線SKIPIF1<0對稱，則SKIPIF1<0．（2）若函數(shù)SKIPIF1<0關(guān)于點SKIPIF1<0對稱，則SKIPIF1<0．（3）函數(shù)SKIPIF1<0與SKIPIF1<0關(guān)于SKIPIF1<0軸對稱，函數(shù)SKIPIF1<0與SKIPIF1<0關(guān)于原點對稱．二、題型分類精講二、題型分類精講真題刷刷刷真題刷刷刷一、單選題1．下列函數(shù)中是增函數(shù)的為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．設(shè)函數(shù)SKIPIF1<0，則下列函數(shù)中為奇函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．設(shè)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<04．已知函數(shù)SKIPIF1<0，則圖象為如圖的函數(shù)可能是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．如圖是下列四個函數(shù)中的某個函數(shù)在區(qū)間SKIPIF1<0的大致圖像，則該函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．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<07．已知函數(shù)SKIPIF1<0的定義域為R，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．0 D．18．已知函數(shù)SKIPIF1<0的定義域均為R，且SKIPIF1<0．若SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對稱，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<09．設(shè)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(dāng)SKIPIF1<0時，SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題10．已知函數(shù)SKIPIF1<0及其導(dǎo)函數(shù)SKIPIF1<0的定義域均為SKIPIF1<0，記SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0均為偶函數(shù)，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0三、填空題11．寫出一個同時具有下列性質(zhì)①②③的函數(shù)SKIPIF1<0_______．①SKIPIF1<0；②當(dāng)SKIPIF1<0時，SKIPIF1<0；③SKIPIF1<0是奇函數(shù)．四、雙空題12．若SKIPIF1<0是奇函數(shù)，則SKIPIF1<0_____，SKIPIF1<0______．題型一函數(shù)的奇偶性策略方法判斷函數(shù)奇偶性的方法(1)定義法：(2)圖象法：(3)性質(zhì)法：在公共定義域內(nèi)有：奇±奇＝奇，偶±偶＝偶，奇×奇＝偶，偶×偶＝偶，奇×偶＝奇．【典例1】判斷下列函數(shù)的奇偶性：(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0；(4)SKIPIF1<0．【題型訓(xùn)練】一、單選題1．函數(shù)SKIPIF1<0的奇偶性是（）A．是奇函數(shù)，不是偶函數(shù)B．是偶函數(shù)，不是奇函數(shù)C．既是奇函數(shù)，也是偶函數(shù)D．非奇非偶函數(shù)2．已知奇函數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時，SKIPIF1<0，則當(dāng)SKIPIF1<0時，SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．若函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0（

）A．2 B．1 C．0 D．SKIPIF1<04．函數(shù)SKIPIF1<0的部分圖象大致為（

）A． B．C． D．二、填空題5．函數(shù)SKIPIF1<0為偶函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0時，SKIPIF1<0___________．6．SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0__________.7．已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0的解集是__________.三、解答題8．已知函數(shù)SKIPIF1<0(1)求函數(shù)SKIPIF1<0解析式；(2)判斷函數(shù)SKIPIF1<0的奇偶性并加以證明9．已知函數(shù)SKIPIF1<0．(1)求SKIPIF1<0的值；(2)令SKIPIF1<0，求證：SKIPIF1<0為奇函數(shù)；(3)若銳角SKIPIF1<0滿足SKIPIF1<0，求SKIPIF1<0的取值范圍．題型二函數(shù)奇偶性的應(yīng)用策略方法已知函數(shù)奇偶性可以解決的三個問題【典例1】若函數(shù)SKIPIF1<0是定義在R上的奇函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．5 D．7【典例2】若函數(shù)SKIPIF1<0是偶函數(shù)，則SKIPIF1<0、SKIPIF1<0的值是（

）A．SKIPIF1<0 B．SKIPIF1<0不能確定，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0不能確定 D．SKIPIF1<0【典例3】偶函數(shù)SKIPIF1<0滿足：SKIPIF1<0，且在區(qū)間SKIPIF1<0與SKIPIF1<0上分別遞減和遞增，使SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0一、單選題1．若函數(shù)SKIPIF1<0為奇函數(shù)，則實數(shù)SKIPIF1<0的值為（

）A．1 B．2 C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0為偶函數(shù)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．已知函數(shù)SKIPIF1<0為SKIPIF1<0上的奇函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0＋1 D．SKIPIF1<04．定義在SKIPIF1<0上的偶函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)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<05．已知偶函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，則SKIPIF1<0的解集是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，且SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題7．已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是偶函數(shù)，在區(qū)間SKIPIF1<0上是單調(diào)函數(shù)，且SKIPIF1<0，則（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．已知函數(shù)SKIPIF1<0的定義域為R，SKIPIF1<0為奇函數(shù)，且對SKIPIF1<0，SKIPIF1<0恒成立，則（

）A．SKIPIF1<0為奇函數(shù) B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0三、填空題9．已知函數(shù)SKIPIF1<0（其中SKIPIF1<0是自然對數(shù)的底數(shù)，SKIPIF1<0）是奇函數(shù)，則實數(shù)SKIPIF1<0的值為______.10．已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，且SKIPIF1<0，則不等式SKIPIF1<0的解集為______.11．定義在SKIPIF1<0上的函數(shù)SKIPIF1<0，滿足SKIPIF1<0為偶函數(shù)，SKIPIF1<0為奇函數(shù)，若SKIPIF1<0，則SKIPIF1<0__________.12．）已知函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，若SKIPIF1<0為奇函數(shù)，且SKIPIF1<0，則SKIPIF1<0_________.題型三函數(shù)的周期性策略方法函數(shù)周期性的判斷與應(yīng)用【典例1】若函數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0可以是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】若定義域為SKIPIF1<0的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】已知定義在SKIPIF1<0上的奇函數(shù)，SKIPIF1<0滿足SKIPIF1<0，當(dāng)SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．函數(shù)SKIPIF1<0是定義在R上奇函數(shù)，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．0 B．SKIPIF1<0 C．2 D．12．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0為奇函數(shù)，則SKIPIF1<0（

）A．0 B．1 C．2 D．33．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0的圖像關(guān)于y軸對稱，且周期為3，又SKIPIF1<0，則SKIPIF1<0的值是（

）A．2023 B．2022 C．SKIPIF1<0 D．14．已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0是偶函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．3 C．SKIPIF1<0 D．SKIPIF1<0二、多選題5．已知函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，都有SKIPIF1<0，且SKIPIF1<0，則下列結(jié)論正確的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<06．已知偶函數(shù)SKIPIF1<0滿足SKIPIF1<0，下列說法正確的是（

）A．函數(shù)SKIPIF1<0是以2為周期的周期函數(shù)B．函數(shù)SKIPIF1<0是以4為周期的周期函數(shù)C．函數(shù)SKIPIF1<0為偶函數(shù)D．函數(shù)SKIPIF1<0為偶函數(shù)三、填空題7．SKIPIF1<0是以2為周期的函數(shù)，若SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0________．8．若定義域為SKIPIF1<0的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0________．9．已知定義在實數(shù)集SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時，SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的最小值為__________.四、解答題10．設(shè)SKIPIF1<0是定義在R上的偶函數(shù)，其圖象關(guān)于直線SKIPIF1<0對稱，對任意SKIPIF1<0，SKIPIF1<0，都有SKIPIF1<0，且SKIPIF1<0．(1)求fSKIPIF1<0；(2)證明SKIPIF1<0是周期函數(shù)；(3)記SKIPIF1<0SKIPIF1<0，求SKIPIF1<0．題型四函數(shù)的對稱性策略方法函數(shù)圖象的對稱性的判斷與應(yīng)用【典例1】已知二次函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】函數(shù)SKIPIF1<0在SKIPIF1<0上是增函數(shù)，函數(shù)SKIPIF1<0是偶函數(shù)，則下列結(jié)論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．下列函數(shù)的圖象中，既是軸對稱圖形又是中心對稱的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．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(guān)系是A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．不能確定3．定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0為奇函數(shù)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2022 D．2023二、多選題4．已知函數(shù)SKIPIF1<0，則下列結(jié)論正確的有（

）A．SKIPIF1<0的圖象關(guān)于坐標(biāo)原點對稱 B．SKIPIF1<0的圖象關(guān)于SKIPIF1<0軸對稱C．SKIPIF1<0的最大值為1 D．SKIPIF1<0在定義域上單調(diào)遞減5．設(shè)函數(shù)f（x）的定義域為R，且函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對稱，函數(shù)SKIPIF1<0的圖像關(guān)于點（3，0）對稱，則下列說法正確的是（

）A．4是f（x）的周期 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0三、填空題6．定義在R上的非常數(shù)函數(shù)SKIPIF1<0滿足：SKIPIF1<0，且SKIPIF1<0.請寫出符合條件的一個函數(shù)的解析式SKIPIF1<0______.7．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，若SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對稱，則SKIPIF1<0_________.8．已知二次函數(shù)SKIPIF1<0（a，b為常數(shù)）滿足SKIPIF1<0，且方程SKIPIF1<0有兩等根，SKIPIF1<0在SKIPIF1<0上的最大值為SKIPIF1<0，則SKIPIF1<0的最大值為__________.四、解答題9．我們知道，函數(shù)SKIPIF1<0的圖象關(guān)于坐標(biāo)原點成中心對稱圖形的充要條件是函數(shù)SKIPIF1<0為奇函數(shù)，有同學(xué)發(fā)現(xiàn)可以將其推廣為：函數(shù)SKIPIF1<0的圖象關(guān)于點SKIPIF1<0成中心對稱圖形的充要條件是函數(shù)SKIPIF1<0為奇函數(shù).（1）求函數(shù)SKIPIF1<0圖象的對稱中心；（2）類比上述推廣結(jié)論，寫出“函數(shù)SKIPIF1<0的圖象關(guān)于y軸成軸對稱圖形的充要條件是函數(shù)SKIPIF1<0為偶函數(shù)”的一個推廣結(jié)論.題型五函數(shù)性質(zhì)的綜合應(yīng)用【典例1】若SKIPIF1<0的定義域為SKIPIF1<0，且滿足SKIPIF1<0為偶函數(shù)，SKIPIF1<0的圖象關(guān)于SKIPIF1<0成中心對稱，則下列說法正確的個數(shù)是（）①SKIPIF1<0的一個周期為4②SKIPIF1<0

③SKIPIF1<0圖象的一條對稱軸為SKIPIF1<0④SKIPIF1<0A．1 B．2 C．3 D．4【題型訓(xùn)練】一、單選題1．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)單調(diào)且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．5055C．SKIPIF1<0 D．10112．定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0為偶函數(shù)，且當(dāng)SKIPIF1<0時，SKIPIF1<0，則下列結(jié)論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．函數(shù)SKIPIF1<0的圖像大致為（

）A．B．C．D．4．定義在SKIPIF1<0上函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIP