新高考數(shù)學二輪復習考點講義第五講函數(shù)的單調(diào)性、奇偶性、周期性講義（原卷版）

第五講：函數(shù)的單調(diào)性、奇偶性、周期性【考點梳理】1．增函數(shù)與減函數(shù)一般地，設函數(shù)SKIPIF1<0的定義域為SKIPIF1<0：(1)如果對于定義域SKIPIF1<0內(nèi)某個區(qū)間SKIPIF1<0上的任意兩個自變量的值SKIPIF1<0，SKIPIF1<0，當SKIPIF1<0時，都有SKIPIF1<0，那么就說函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是增函數(shù)．(2)如果對于定義域SKIPIF1<0內(nèi)某個區(qū)間SKIPIF1<0上的任意兩個自變量的值SKIPIF1<0，SKIPIF1<0，當SKIPIF1<0時，都有SKIPIF1<0，那么就說函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是減函數(shù)．2．函數(shù)的最大值與最小值一般地，設函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，如果存在實數(shù)SKIPIF1<0滿足：(1)對于任意的SKIPIF1<0，都有SKIPIF1<0；存在SKIPIF1<0，使得SKIPIF1<0，那么，我們稱SKIPIF1<0是函數(shù)SKIPIF1<0的最大值．(2)對于任意的SKIPIF1<0，都有SKIPIF1<0；存在SKIPIF1<0，使得SKIPIF1<0，那么我們稱SKIPIF1<0是函數(shù)SKIPIF1<0的最小值．3．函數(shù)單調(diào)性的兩個等價結(jié)論設SKIPIF1<0則(1)SKIPIF1<0(或SKIPIF1<0在SKIPIF1<0上單調(diào)遞增。(2)SKIPIF1<0(或SKIPIF1<0?f(x)在SKIPIF1<0上單調(diào)遞減．4．函數(shù)的奇偶性奇偶性定義圖象特點偶函數(shù)如果函數(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是偶函數(shù)關于SKIPIF1<0對稱奇函數(shù)如果函數(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是奇函數(shù)關于原點對稱5.奇偶函數(shù)的性質(zhì)(1)奇函數(shù)在關于原點對稱的區(qū)間上的單調(diào)性相同，偶函數(shù)在關于原點對稱的區(qū)間上的單調(diào)性相反．(2)在公共定義域內(nèi)(ⅰ)兩個奇函數(shù)的和函數(shù)是奇函數(shù)，兩個奇函數(shù)的積函數(shù)是偶函數(shù).(ⅱ)兩個偶函數(shù)的和函數(shù)、積函數(shù)是偶函數(shù).(ⅲ)一個奇函數(shù)與一個偶函數(shù)的積函數(shù)是奇函數(shù).(3)若SKIPIF1<0是奇函數(shù)且SKIPIF1<0處有意義，則SKIPIF1<0.6．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)，那么這個最小正數(shù)就叫做SKIPIF1<0的最小正周期．(3)常見結(jié)論：若SKIPIF1<0，則SKIPIF1<0；若SKIPIF1<0，則SKIPIF1<0；若SKIPIF1<0，則SKIPIF1<0.【典型題型講解】考點一：函數(shù)的單調(diào)性【典例例題】例1.若定義在R上的函數(shù)f(x)對任意兩個不相等的實數(shù)a，b，總有SKIPIF1<0>0成立，則必有（

）A．f(x)在R上是增函數(shù) B．f(x)在R上是減函數(shù)C．函數(shù)f(x)先增后減 D．函數(shù)f(x)先減后增【方法技巧與總結(jié)】函數(shù)單調(diào)性的判斷方法①定義法：根據(jù)增函數(shù)、減函數(shù)的定義，按照“取值—變形—判斷符號—下結(jié)論”進行判斷．②圖象法：就是畫出函數(shù)的圖象，根據(jù)圖象的上升或下降趨勢，判斷函數(shù)的單調(diào)性．③直接法：就是對我們所熟悉的函數(shù)，如一次函數(shù)、二次函數(shù)、反比例函數(shù)等，直接寫出它們的單調(diào)區(qū)間．【變式訓練】1.已知函數(shù)SKIPIF1<0，若SKIPIF1<0，則實數(shù)SKIPIF1<0的取值范圍是___.2.已知函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，且對任意兩個不相等的實數(shù)SKIPIF1<0，SKIPIF1<0都有SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·廣東惠州·一模）已知SKIPIF1<0，則當SKIPIF1<0時，SKIPIF1<0與SKIPIF1<0的大小關系是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．不確定4.“SKIPIF1<0”是“函數(shù)SKIPIF1<0是在SKIPIF1<0上的單調(diào)函數(shù)”的（

）A．充分不必要條件 B．必要不充分條件C．充要條件 D．既不充分也不必要條件5．已知函數(shù)SKIPIF1<0若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0僅有1個零點，則實數(shù)m的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．若函數(shù)SKIPIF1<0是SKIPIF1<0上的單調(diào)函數(shù)，則SKIPIF1<0的取值范圍（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0考點二：判斷函數(shù)的奇偶性【典例例題】例1.已知函數(shù)SKIPIF1<0，則SKIPIF1<0（

）A．是偶函數(shù)，且在SKIPIF1<0是單調(diào)遞增 B．是奇函數(shù)，且在SKIPIF1<0是單調(diào)遞增C．是偶函數(shù)，且在SKIPIF1<0是單調(diào)遞減 D．是奇函數(shù)，且在SKIPIF1<0是單調(diào)遞減【方法技巧與總結(jié)】函數(shù)的奇偶性的判斷：圖像法、解析式法；常見函數(shù)的奇偶性?！咀兪接柧殹?．下列函數(shù)中，在其定義域內(nèi)既是奇函數(shù)又是減函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·廣東·二模）存在函數(shù)SKIPIF1<0使得對于SKIPIF1<0都有SKIPIF1<0，則函數(shù)SKIPIF1<0可能為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·廣東湛江·一模）下列函數(shù)是奇函數(shù)，且函數(shù)值恒小于1的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．（2022·廣東廣東·一模）下列四個函數(shù)中，以SKIPIF1<0為周期且在SKIPIF1<0上單調(diào)遞增的偶函數(shù)有（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0考點三：函數(shù)的奇偶性的應用【典例例題】例1．（2022·廣東中山·高三期末）（多選）已知函數(shù)SKIPIF1<0，則下列說法正確的是（

）A．函數(shù)SKIPIF1<0是偶函數(shù) B．函數(shù)SKIPIF1<0是奇函數(shù)C．函數(shù)SKIPIF1<0在SKIPIF1<0上為增函數(shù) D．函數(shù)SKIPIF1<0的值域為SKIPIF1<0例2．（2022·廣東汕尾·高三期末）我國著名數(shù)學家華羅庚先生曾說：數(shù)缺形時少直觀，形缺數(shù)時難入微，數(shù)形結(jié)合百般好，隔裂分家萬事休，在數(shù)學的學習和研究中，函數(shù)的解析式常用來研究函數(shù)圖象的特征，函數(shù)SKIPIF1<0的圖象大致為（

）A． B．C． D．【方法技巧與總結(jié)】函數(shù)單調(diào)性與奇偶性結(jié)合時，注意函數(shù)單調(diào)性和奇偶性的定義，以及奇偶函數(shù)圖像的對稱性.【變式訓練】1.（2021·廣東汕頭·高三期末）已知偶函數(shù)f(x)在區(qū)間SKIPIF1<0上單調(diào)遞減，若f(-1)=0，則滿足f(m)>0的實數(shù)m的取值范圍是______．2．（2022·廣東·金山中學高三期末）已知函數(shù)SKIPIF1<0，則SKIPIF1<0________．3．（2022·廣東深圳·一模）已知函數(shù)SKIPIF1<0是定義域為R的奇函數(shù)，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0_________．4．（2022·廣東韶關·一模）已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<05．（2022·廣東·一模）已知函數(shù)SKIPIF1<0，SKIPIF1<0，則圖象如圖的函數(shù)可能是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022·廣東湛江·一模）下列函數(shù)是奇函數(shù)，且函數(shù)值恒小于1的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．（2022·廣東廣州·一模）若函數(shù)SKIPIF1<0的大致圖象如圖，則SKIPIF1<0的解析式可能是(

)SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2022·廣東廣東·一模）函數(shù)SKIPIF1<0的部分圖象大致為（

）A． B．C． D．考點四：函數(shù)的對稱性和周期性【典例例題】例1．設函數(shù)SKIPIF1<0的定義域為D，若對任意的SKIPIF1<0，且SKIPIF1<0，恒有SKIPIF1<0，則稱函數(shù)SKIPIF1<0具有對稱性，其中點SKIPIF1<0為函數(shù)SKIPIF1<0的對稱中心，研究函數(shù)SKIPIF1<0的對稱中心，求SKIPIF1<0（

）A．2022 B．4043 C．4044 D．8086【方法技巧與總結(jié)】（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.【變式訓練】1．（2022·廣東珠海·高三期末）已知SKIPIF1<0是定義域在SKIPIF1<0上的奇函數(shù)，且滿足SKIPIF1<0．當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<02．已知定義在R上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0是奇函數(shù)，則（

）A．SKIPIF1<0是偶函數(shù) B．SKIPIF1<0的圖象關于直線SKIPIF1<0對稱C．SKIPIF1<0是奇函數(shù) D．SKIPIF1<0的圖象關于點SKIPIF1<0對稱3．已知函數(shù)SKIPIF1<0的定義域為R，且SKIPIF1<0對任意SKIPIF1<0恒成立，又函數(shù)SKIPIF1<0的圖象關于點SKIPIF1<0對稱，且SKIPIF1<0，則SKIPIF1<0（

）A．2021 B．SKIPIF1<0 C．2022 D．SKIPIF1<04．已知定義在R上的偶函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0，則下面結(jié)論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．已知函數(shù)SKIPIF1<0滿足SKIPIF1<0對任意SKIPIF1<0恒成立，又函數(shù)SKIPIF1<0的圖象關于點SKIPIF1<0對稱，且SKIPIF1<0則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知函數(shù)SKIPIF1<0是SKIPIF1<0上的奇函數(shù)，且SKIPIF1<0，且當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知SKIPIF1<0是定義在R上的奇函數(shù)，若SKIPIF1<0為偶函數(shù)且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．6【鞏固練習】一、單選題1．下列函數(shù)中，在其定義域內(nèi)既是奇函數(shù)又是減函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0，不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0的最大值是M，最小值是m，則SKIPIF1<0的值等于(

)A．0 B．10 C．SKIPIF1<0 D．SKIPIF1<04．已知函數(shù)SKIPIF1<0的圖象關于原點對稱，且SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0（

）A．-11 B．-8 C．SKIPIF1<0 D．SKIPIF1<0二、多選題5．下面關于函數(shù)SKIPIF1<0的性質(zhì)，說法正確的是（

）A．SKIPIF1<0的定義域為SKIPIF1<0 B．SKIPIF1<0的值域為SKIPIF1<0C．SKIPIF1<0在定義域上單調(diào)遞減 D．點SKIPIF1<0是SKIPIF1<0圖象的對稱中心6．已知定義在R上的偶函數(shù)SKIPIF1<0的圖像是連續(xù)的，SKIPIF1<0，SKIPIF1<0在區(qū)間SKIPIF1<0上是增函數(shù)，則下列結(jié)論正確的是（

）A．SKIPIF1<0的一個周期為6 B．SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減C．SKIPIF1<0的圖像關于直線SKIPIF1<0對稱 D．SKIPIF1<0在區(qū)間SKIPIF1<0上共有100個零點7．（2022·重慶巴蜀中學高三階段練習）已知函數(shù)SKIPIF1<0對任意SKIPIF1<0都有SKIPIF1<0，若函數(shù)SKIPIF1<0的圖象關于SKIPIF1<0對稱，且對任意的SKIPIF1<0，且SKIPIF1<0，都有SKIP