新高考數(shù)學(xué)二輪復(fù)習(xí)專題2-1 函數(shù)性質(zhì)（單調(diào)性、奇偶性、中心對(duì)稱、軸對(duì)稱、周期性）（原卷版）

專題2-1函數(shù)性質(zhì)（單調(diào)性、奇偶性、中心對(duì)稱、軸對(duì)稱、周期性）目錄TOC\o"1-1"\h\u題型01奇偶性基礎(chǔ) 1題型02中心對(duì)稱型函數(shù) 2題型03軸對(duì)稱型函數(shù) 3題型04斜直線軸對(duì)稱型 3題型05“正余弦”型對(duì)稱 4題型06伸縮型對(duì)稱 5題型07一元三次函數(shù)型中心對(duì)稱 6題型08“局部周期”型函數(shù)性質(zhì) 7題型09雙函數(shù)型對(duì)稱 8題型10原函數(shù)與導(dǎo)函數(shù)型雙函數(shù)對(duì)稱 9題型11放大鏡型函數(shù)性質(zhì) 10題型12抽象函數(shù)賦值型性質(zhì) 11題型13對(duì)稱型恒成立求參 11題型14構(gòu)造“對(duì)稱”型函數(shù) 12高考練場(chǎng) 13題型01奇偶性基礎(chǔ)【解題攻略】奇偶函數(shù)的性質(zhì)①偶函數(shù)?f(－x)＝f(x)?關(guān)于y軸對(duì)稱?對(duì)稱區(qū)間的單調(diào)性相反；②奇函數(shù)?f(－x)＝－f(x)?關(guān)于原點(diǎn)對(duì)稱?對(duì)稱區(qū)間的單調(diào)性相同；③奇函數(shù)在x＝0處有意義時(shí)，必有結(jié)論f(0)＝0；奇偶性的判定①“奇±奇”是奇，“偶±偶”是偶，“奇×/÷奇”是偶，“偶×/÷偶”是偶，“奇×/÷偶”是奇；②奇(偶)函數(shù)倒數(shù)或相反數(shù)運(yùn)算，奇偶性不變； ③奇(偶)函數(shù)的絕對(duì)值運(yùn)算，函數(shù)的奇偶性均為偶函數(shù)．【典例1-1】（2023秋·山西·高三校聯(lián)考期中）已知函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0的值是（

）A．0 B．SKIPIF1<0 C．12 D．10【典例1-2】（2023秋·北京昌平·高三北京市昌平區(qū)前鋒學(xué)校?？茧A段練習(xí)）已知SKIPIF1<0，則（

）A．SKIPIF1<0為偶函數(shù)，且在SKIPIF1<0上單調(diào)遞增B．SKIPIF1<0為偶函數(shù)，且在SKIPIF1<0上單調(diào)遞減C．SKIPIF1<0為奇函數(shù)，且在SKIPIF1<0上單調(diào)遞增D．SKIPIF1<0為奇函數(shù)，且在SKIPIF1<0上單調(diào)遞減【變式1-1】．（2023·全國(guó)·高一專題練習(xí)）若SKIPIF1<0為奇函數(shù)，則SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1-2】（2023秋·江蘇南通·高三統(tǒng)考開學(xué)考試）已知SKIPIF1<0是奇函數(shù)，則SKIPIF1<0在SKIPIF1<0處的切線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1-3】．（2023秋·天津和平·高三天津一中?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0，若對(duì)任意SKIPIF1<0，都有SKIPIF1<0成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型02中心對(duì)稱型函數(shù)【解題攻略】中心對(duì)稱結(jié)論：（1）若函數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0的一個(gè)對(duì)稱中心為SKIPIF1<0（2）若函數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0的一個(gè)對(duì)稱中心為SKIPIF1<0（3）若函數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0的一個(gè)對(duì)稱中心為SKIPIF1<0.【典例1-1】已知函數(shù)SKIPIF1<0，則存在非零實(shí)數(shù)SKIPIF1<0，使得（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例1-2】函數(shù)SKIPIF1<0的圖象與函數(shù)SKIPIF1<0圖象的所有交點(diǎn)的橫坐標(biāo)之和為___________.【變式1-1】.設(shè)函數(shù)SKIPIF1<0的最大值為5，則SKIPIF1<0的最小值為（）A．SKIPIF1<0 B．1 C．2 D．3【變式1-2】已知函數(shù)SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0使關(guān)于SKIPIF1<0的不等式SKIPIF1<0成立，則實(shí)數(shù)SKIPIF1<0的范圍為___________.【變式1-3】.函數(shù)SKIPIF1<0的圖像可能是（）A． B．C． D．題型03軸對(duì)稱型函數(shù)【解題攻略】軸對(duì)稱性的常用結(jié)論如下：若函數(shù)SKIPIF1<0滿足SKIPIF1<0,則SKIPIF1<0的一條對(duì)稱軸為SKIPIF1<0若函數(shù)SKIPIF1<0滿足SKIPIF1<0,則SKIPIF1<0的一條對(duì)稱軸為SKIPIF1<0若函數(shù)SKIPIF1<0滿足SKIPIF1<0,則SKIPIF1<0的一條對(duì)稱軸為SKIPIF1<0(4)f(a－x)=f(b＋x)?f(x)的圖象關(guān)于直線x=eq\f(a＋b,2)對(duì)稱；【典例1-1】．（2023上·重慶·高三重慶市忠縣忠州中學(xué)校校聯(lián)考）已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0，函數(shù)SKIPIF1<0為偶函數(shù)，且對(duì)SKIPIF1<0都有SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的取值范圍是．【典例1-2】（2023上·江西景德鎮(zhèn)·高一統(tǒng)考期中）已知函數(shù)SKIPIF1<0滿足關(guān)系式SKIPIF1<0，且對(duì)于SKIPIF1<0，SKIPIF1<0，滿足SKIPIF1<0恒成立，若不等式SKIPIF1<0對(duì)SKIPIF1<0恒成立，則實(shí)數(shù)a的取值范圍是．【變式1-1】．（2023上·江蘇南通·高三統(tǒng)考階段練習(xí)）設(shè)定義在SKIPIF1<0上的函數(shù)SKIPIF1<0在SKIPIF1<0單調(diào)遞減，且SKIPIF1<0為偶函數(shù)，若SKIPIF1<0，SKIPIF1<0，且有SKIPIF1<0，則SKIPIF1<0的最小值為.【變式1-2】（2023上·山東濟(jì)南·高三統(tǒng)考開學(xué)考試）若函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，且SKIPIF1<0有且僅有4個(gè)零點(diǎn)，則SKIPIF1<0的值為．【變式1-3】．（2023上·陜西榆林·高三?？茧A段練習(xí)）函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且圖象關(guān)于SKIPIF1<0對(duì)稱，在區(qū)間SKIPIF1<0上，SKIPIF1<0，則SKIPIF1<0.題型04斜直線軸對(duì)稱型【解題攻略】關(guān)于斜直線軸對(duì)稱，可以借鑒圓錐曲線中直線的對(duì)稱性來處理（1）點(diǎn)SKIPIF1<0關(guān)于直線SKIPIF1<0的對(duì)稱點(diǎn)SKIPIF1<0，則有SKIPIF1<0；（2）直線關(guān)于直線的對(duì)稱可轉(zhuǎn)化為點(diǎn)關(guān)于直線的對(duì)稱問題來解決.如果斜直線軸對(duì)稱，還有以下經(jīng)驗(yàn)公式：如果對(duì)稱軸所在的直線斜率是SKIPIF1<0，即直線是SKIPIF1<0型，可以利用反解對(duì)稱軸法直接求出對(duì)稱變換式子SKIPIF1<0(1)如果SKIPIF1<0關(guān)于直線SKIPIF1<0的對(duì)稱點(diǎn)為SKIPIF1<0，則SKIPIF1<0的坐標(biāo)為SKIPIF1<0；(2)如果SKIPIF1<0關(guān)于直線SKIPIF1<0的對(duì)稱點(diǎn)為SKIPIF1<0，則SKIPIF1<0的坐標(biāo)為SKIPIF1<0．【典例1-1】（2023上·重慶·高三西南大學(xué)附中校考）已知函數(shù)SKIPIF1<0為奇函數(shù)，SKIPIF1<0的函數(shù)圖象關(guān)于SKIPIF1<0對(duì)稱，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0.【典例1-2】（2023上·遼寧·高三校聯(lián)考）已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且其圖象關(guān)于直線SKIPIF1<0對(duì)稱，若當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0.【變式1-1】（2023上·遼寧大連·高三大連八中?？计谥校┮阎瘮?shù)SKIPIF1<0，若曲線SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱，則SKIPIF1<0的值為．【變式1-2】（2023上·上海浦東新·高三華師大二附中?？迹┮阎瘮?shù)SKIPIF1<0的圖象過點(diǎn)SKIPIF1<0，且關(guān)于直線SKIPIF1<0成軸對(duì)稱圖形，則SKIPIF1<0．【變式1-3】（2021上·高一?？颊n時(shí)練習(xí)）若函數(shù)SKIPIF1<0的圖象與SKIPIF1<0且SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，則SKIPIF1<0的值等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型05“正余弦”型對(duì)稱【解題攻略】SKIPIF1<0（1）兩中心SKIPIF1<0；（2）兩垂直軸SKIPIF1<0則SKIPIF1<0；（3）一個(gè)中心SKIPIF1<0，一條軸SKIPIF1<0，則SKIPIF1<0【典例1-1】函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且SKIPIF1<0為偶函數(shù)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，若函數(shù)SKIPIF1<0恰有一個(gè)零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值集合是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例1-2】.定義在SKIPIF1<0上的偶函數(shù)f（x）滿足f（－x）＋f（x－2）＝0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0（已知SKIPIF1<0），則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式1-1】已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足條件SKIPIF1<0，且函數(shù)SKIPIF1<0為奇函數(shù)，則下列說法中錯(cuò)誤的是（

）A．函數(shù)SKIPIF1<0是周期函數(shù)；B．函數(shù)SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱；C．函數(shù)SKIPIF1<0為SKIPIF1<0上的偶函數(shù)；D．函數(shù)SKIPIF1<0為SKIPIF1<0上的單調(diào)函數(shù)．【變式1-2】已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，SKIPIF1<0為SKIPIF1<0的導(dǎo)函數(shù)，且SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0為偶函數(shù)，則下列結(jié)論不一定成立的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式1-3】.定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0；且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．則方程SKIPIF1<0所有的根之和為（

）A．6 B．12 C．14 D．10題型06伸縮型對(duì)稱【解題攻略】伸縮變換y＝f(ax)y＝f(x)eq\o(→,\s\up7(a>1，縱坐標(biāo)伸長(zhǎng)為原來的a倍，橫坐標(biāo)不變),\s\do5(0<a<1，縱坐標(biāo)縮短為原來的a倍，橫坐標(biāo)不變))y＝af(x)【典例1-1】（2023秋·湖南懷化·高三統(tǒng)考）已知SKIPIF1<0不是常函數(shù)，且是定義域?yàn)镾KIPIF1<0的奇函數(shù)，若SKIPIF1<0的最小正周期為1，則（

）A．SKIPIF1<0 B．1是SKIPIF1<0的一個(gè)周期C．SKIPIF1<0 D．SKIPIF1<0【典例1-2】（2023·河南·長(zhǎng)葛市第一高級(jí)中學(xué)統(tǒng)考模擬預(yù)測(cè)）若函數(shù)f（x）的定義域?yàn)镽，且f（2x＋1）為偶函數(shù)，f（x－1）的圖象關(guān)于點(diǎn)（3，3）成中心對(duì)稱，則下列說法正確的個(gè)數(shù)為（

）①SKIPIF1<0的一個(gè)周期為2

②SKIPIF1<0③SKIPIF1<0④直線SKIPIF1<0是SKIPIF1<0圖象的一條對(duì)稱軸A．1 B．2 C．3 D．4【變式1-1】（2022秋·重慶南岸·高三重慶市第十一中學(xué)校?？茧A段練習(xí)）已知SKIPIF1<0是定義在SKIPIF1<0上的函數(shù)，SKIPIF1<0是奇函數(shù)，且SKIPIF1<0是偶函數(shù)，則下列選項(xiàng)一定正確的是（

）A．函數(shù)SKIPIF1<0的周期為2 B．函數(shù)SKIPIF1<0的周期為3C．SKIPIF1<0 D．SKIPIF1<0【變式1-2】．（2022秋·吉林長(zhǎng)春·高三長(zhǎng)春市第二中學(xué)校考階段練習(xí)）設(shè)函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，且SKIPIF1<0是奇函數(shù)，SKIPIF1<0是偶函數(shù)，則一定有（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1-3】（2022秋·廣西玉林·高三校聯(lián)考階段練習(xí)）已知SKIPIF1<0是定義域?yàn)镾KIPIF1<0的奇函數(shù)，SKIPIF1<0是定義域?yàn)镾KIPIF1<0的偶函數(shù)，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型07一元三次函數(shù)型中心對(duì)稱【解題攻略】所有的三次函數(shù)SKIPIF1<0都有“拐點(diǎn)”，且該“拐點(diǎn)”也是函數(shù)SKIPIF1<0的圖像的對(duì)稱中心，設(shè)SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)數(shù)，SKIPIF1<0是SKIPIF1<0的導(dǎo)數(shù)，若方程SKIPIF1<0有實(shí)數(shù)解SKIPIF1<0，則稱點(diǎn)SKIPIF1<0為函數(shù)SKIPIF1<0的“拐點(diǎn)”．【典例1-1】．給出定義：設(shè)SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，若方程SKIPIF1<0有實(shí)數(shù)解SKIPIF1<0，則稱SKIPIF1<0為函數(shù)SKIPIF1<0的“拐點(diǎn)”.經(jīng)研究發(fā)現(xiàn)所有的三次函數(shù)SKIPIF1<0都有“拐點(diǎn)”，且該“拐點(diǎn)”也是函數(shù)SKIPIF1<0的圖像的對(duì)稱中心，若函數(shù)SKIPIF1<0，則SKIPIF1<0（

）A．8082 B．2021 C．-8082 D．-2023【典例1-2】已知一元三次函數(shù)對(duì)稱中心的橫坐標(biāo)為其二階導(dǎo)函數(shù)的零點(diǎn)．若SKIPIF1<0，則SKIPIF1<0（

）A．0 B．4 C．SKIPIF1<0 D．SKIPIF1<0【變式1-1】在同一坐標(biāo)系中作出三次函數(shù)SKIPIF1<0及其導(dǎo)函數(shù)的圖象，下列可能正確的序號(hào)是（

）A．①② B．①③ C．③④ D．①④【變式1-2】設(shè)函數(shù)SKIPIF1<0是SKIPIF1<0的導(dǎo)數(shù)，經(jīng)過探究發(fā)現(xiàn)，任意一個(gè)三次函數(shù)SKIPIF1<0SKIPIF1<0的圖象都有對(duì)稱中心SKIPIF1<0，其中SKIPIF1<0滿足SKIPIF1<0，已知函數(shù)SKIPIF1<0，則SKIPIF1<0（

）A．0 B．SKIPIF1<0 C．1 D．SKIPIF1<0【變式1-3】一般地，對(duì)于一元三次函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0為三次函數(shù)SKIPIF1<0的對(duì)稱中心，已知函數(shù)SKIPIF1<0圖象的對(duì)稱中心的橫坐標(biāo)為SKIPIF1<0，且SKIPIF1<0有三個(gè)零點(diǎn)，則實(shí)數(shù)a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型08“局部周期”型函數(shù)性質(zhì)【解題攻略】局部周期函數(shù)，可類比以下函數(shù)圖像：SKIPIF1<0【典例1-1】定義在0,+∞上的函數(shù)fx滿足f（i）f2021（ii）若方程fx?kx=0有且只有兩個(gè)解，則實(shí)數(shù)福建省長(zhǎng)汀縣第一中學(xué)2022屆高三上學(xué)期第二次月考數(shù)學(xué)試題【典例1-2】.已知fx=12x+a,x≤0,fx?1【變式1-1】（2021下·天津武清·高三天津市武清區(qū)楊村第一中學(xué)校）已知函數(shù)SKIPIF1<0，若對(duì)于正數(shù)SKIPIF1<0，直線SKIPIF1<0與函數(shù)SKIPIF1<0的圖像恰好有SKIPIF1<0個(gè)不同的交點(diǎn)，則SKIPIF1<0.【變式1-2】．（2021上·四川資陽·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0，函數(shù)SKIPIF1<0在SKIPIF1<0處的切線為SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0與SKIPIF1<0的圖象的公共點(diǎn)個(gè)數(shù)為．題型09雙函數(shù)型對(duì)稱【解題攻略】雙函數(shù)性質(zhì)：1.雙函數(shù)各自對(duì)應(yīng)的對(duì)稱中心和對(duì)稱軸等性質(zhì)2.雙函數(shù)之間存在著互相轉(zhuǎn)化或者互相表示的函數(shù)等量關(guān)系【典例1-1】（2023·廣西玉林·統(tǒng)考模擬預(yù)測(cè)）已知函數(shù)SKIPIF1<0，SKIPIF1<0的定義域均為SKIPIF1<0，SKIPIF1<0是奇函數(shù)，且SKIPIF1<0，SKIPIF1<0，則（

）A．f（x）為奇函數(shù) B．g（x）為奇函數(shù)C．SKIPIF1<0 D．SKIPIF1<0【典例1-2】（2023春·河南開封·高三統(tǒng)考開學(xué)考試）已知函數(shù)SKIPIF1<0，SKIPIF1<0的定義域?yàn)镾KIPIF1<0，且SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0為偶函數(shù)．SKIPIF1<0，則SKIPIF1<0（

）A．24 B．26 C．28 D．30【變式1-1】（2023秋·江西·高三校聯(lián)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0的定義域均為SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0.若SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，且SKIPIF1<0，則SKIPIF1<0（

）A．80 B．86 C．90 D．96【變式1-2】（2023秋·全國(guó)·高三校聯(lián)考階段練習(xí)）SKIPIF1<0的定義域?yàn)镾KIPIF1<0，SKIPIF1<0為偶函數(shù)，SKIPIF1<0且SKIPIF1<0，則下列說法不正確的是（

）A．SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱 B．SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱C．4為SKIPIF1<0的周期 D．SKIPIF1<0【變式1-3】（2022秋·四川成都·高三成都七中?？紝ｎ}練習(xí)）已知函數(shù)SKIPIF1<0的定義域均為SKIPIF1<0為偶函數(shù)，且SKIPIF1<0，SKIPIF1<0，下列說法正確的有（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱B．函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱C．函數(shù)SKIPIF1<0是以4為周期的周期函數(shù)D．函數(shù)SKIPIF1<0是以6為周期的周期函數(shù)題型10原函數(shù)與導(dǎo)函數(shù)型雙函數(shù)對(duì)稱【解題攻略】原函數(shù)與導(dǎo)函數(shù)的性質(zhì)性質(zhì)1若函數(shù)SKIPIF1<0是可導(dǎo)函數(shù)，且圖像關(guān)于SKIPIF1<0對(duì)稱，則其導(dǎo)函數(shù)SKIPIF1<0的圖像關(guān)于SKIPIF1<0軸對(duì)稱性質(zhì)2奇函數(shù)的導(dǎo)數(shù)為偶函數(shù)性質(zhì)3若函數(shù)SKIPIF1<0是可導(dǎo)函數(shù)，且圖像關(guān)于SKIPIF1<0對(duì)稱，則其導(dǎo)函數(shù)SKIPIF1<0的圖像關(guān)于SKIPIF1<0軸對(duì)稱性質(zhì)4偶函數(shù)的導(dǎo)數(shù)為奇函數(shù)性質(zhì)5若函數(shù)SKIPIF1<0是可導(dǎo)函數(shù)，且圖像關(guān)于SKIPIF1<0對(duì)稱，則其導(dǎo)函數(shù)SKIPIF1<0的圖像關(guān)于SKIPIF1<0對(duì)稱偶函數(shù)的導(dǎo)數(shù)為奇函數(shù)性質(zhì)6若定義在R上的函數(shù)SKIPIF1<0是可導(dǎo)函數(shù)，且周期為T，則其導(dǎo)函數(shù)SKIPIF1<0是周期函數(shù)，且周期也為T性質(zhì)7若函數(shù)SKIPIF1<0是可導(dǎo)函數(shù)，定義域?yàn)镈，其導(dǎo)函數(shù)SKIPIF1<0的圖像關(guān)于SKIPIF1<0軸對(duì)稱，則SKIPIF1<0圖像關(guān)于SKIPIF1<0對(duì)稱，SKIPIF1<0為定義域內(nèi)任意一點(diǎn)【典例1-1】（2023·四川成都·校聯(lián)考模擬預(yù)測(cè)）已知函數(shù)SKIPIF1<0及其導(dǎo)函數(shù)SKIPIF1<0的定義域均為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0是偶函數(shù)，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．2022 B．2023 C．2024 D．2025【典例1-2】（2022上·四川遂寧·高三射洪中學(xué)校考階段練習(xí)）已知函數(shù)SKIPIF1<0及其導(dǎo)函數(shù)SKIPIF1<0定義域均為SKIPIF1<0，SKIPIF1<0為奇函數(shù)，SKIPIF1<0，SKIPIF1<0，則正確的有（

）①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0.A．①④ B．①② C．②③ D．③④【變式1-1】（2023·廣西梧州·蒼梧中學(xué)校考模擬預(yù)測(cè)）設(shè)定義在SKIPIF1<0上的函數(shù)SKIPIF1<0與SKIPIF1<0的導(dǎo)函數(shù)分別為SKIPIF1<0和SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0為奇函數(shù)，SKIPIF1<0．現(xiàn)有下列四個(gè)結(jié)論：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0．其中所有正確結(jié)論的序號(hào)是（

）A．①②③ B．②③④ C．①③④ D．①②④【變式1-2】（2023·全國(guó)·高三專題練習(xí)）設(shè)定義在R上的函數(shù)SKIPIF1<0與SKIPIF1<0的導(dǎo)函數(shù)分別為SKIPIF1<0和SKIPIF1<0．若SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0為奇函數(shù)，則下列說法中一定正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0【變式1-3】7.設(shè)定義在實(shí)數(shù)集SKIPIF1<0上的函數(shù)SKIPIF1<0與SKIPIF1<0的導(dǎo)數(shù)分別為SKIPIF1<0與SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0為奇函數(shù)，則下列說法不正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0圖象關(guān)于直線SKIPIF1<0對(duì)稱C．SKIPIF1<0 D．SKIPIF1<0遼寧省沈陽市第二中學(xué)2022-2023學(xué)年高三上學(xué)期12月月考數(shù)學(xué)試題題型11放大鏡型函數(shù)性質(zhì)【解題攻略】形如SKIPIF1<0等“似周期函數(shù)”或者“類周期函數(shù)”，俗稱放大鏡函數(shù)，要注意以下幾點(diǎn)辨析：1.是從左往右放大，還是從右往左放大。2.放大（縮?。r(shí)，要注意是否函數(shù)值有0。3.放大（縮?。r(shí)，是否發(fā)生了上下平移。4.“放大鏡”函數(shù)，在尋找“切線”型臨界值時(shí)，計(jì)算容易“卡殼”，授課時(shí)要著重講清此處計(jì)算。【典例1-1】定義在SKIPIF1<0上函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則使得SKIPIF1<0在SKIPIF1<0上恒成立的SKIPIF1<0的最小值是______________.【典例1-2】.已知SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0有下列結(jié)論：①函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增；②函數(shù)SKIPIF1<0的圖象與直線SKIPIF1<0有且僅有SKIPIF1<0個(gè)不同的交點(diǎn)；③若關(guān)于SKIPIF1<0的方程SKIPIF1<0恰有SKIPIF1<0個(gè)不相等的實(shí)數(shù)根，則這SKIPIF1<0個(gè)實(shí)數(shù)根之和為SKIPIF1<0；④記函數(shù)SKIPIF1<0在SKIPIF1<0上的最大值為SKIPIF1<0，則數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0.其中所有正確結(jié)論的編號(hào)是___________.【變式1-1】已知定義在[1,+∞)上的函數(shù)f(x)=4?A．在[1,6]上，方程f(x)?16x=0B．關(guān)于x的方程f(x)?12nC．當(dāng)x∈[2n?1,2n](n∈D．對(duì)于實(shí)數(shù)x∈[1,+∞)，不等式xf(x)≤6恒成立【變式1-2】設(shè)函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．若對(duì)任意SKIPIF1<0，都有SKIPIF1<0，則m的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式1-3】．定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0滿足：SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，若SKIPIF1<0時(shí)，SKIPIF1<0恒成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型12抽象函數(shù)賦值型性質(zhì)【典例1-1】（2023春·遼寧·高三校聯(lián)考階段練習(xí)）已知SKIPIF1<0是定義在SKIPIF1<0上的函數(shù)，且在區(qū)間SKIPIF1<0內(nèi)單調(diào)遞增，對(duì)SKIPIF1<0，SKIPIF1<0，都有SKIPIF1<0.若SKIPIF1<0，使得不等式SKIPIF1<0成立，則實(shí)數(shù)SKIPIF1<0的最大值為.【典例1-2】．（2023·全國(guó)·高三對(duì)口高考）已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0對(duì)任意實(shí)數(shù)x，y滿足SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0．給出下列結(jié)論：①SKIPIF1<0；②SKIPIF1<0為奇函數(shù)；③SKIPIF1<0為周期函數(shù)；④SKIPIF1<0在SKIPIF1<0內(nèi)單調(diào)遞減．其中正確結(jié)論的序號(hào)是．【變式1-1】（2023·江蘇南通·統(tǒng)考模擬預(yù)測(cè)）若函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0．【變式1-2】（2023·浙江·高三專題練習(xí)）若定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足：SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，則滿足上述條件的函數(shù)SKIPIF1<0可以為.（寫出一個(gè)即可）【變式1-3】（2022秋·湖南衡陽·高三衡陽市一中?？迹┒x在R上的函數(shù)f(x)滿足SKIPIF1<0x，ySKIPIF1<0R，SKIPIF1<0且f(0)SKIPIF1<00，f(a)=0(a>0).則下列結(jié)論正確的序號(hào)有.①f(0)=1；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0.題型13對(duì)稱型恒成立求參【解題攻略】一般地，已知函數(shù)SKIPIF1<0，SKIPIF1<0（1）若SKIPIF1<0，SKIPIF1<0，有SKIPIF1<0成立，故SKIPIF1<0；（2）若SKIPIF1<0，SKIPIF1<0，有SKIPIF1<0成立，故SKIPIF1<0；（3）若SKIPIF1<0，SKIPIF1<0，有SKIPIF1<0成立，故SKIPIF1<0；（4）若SKIPIF1<0，SKIPIF1<0，有SKIPIF1<0，則SKIPIF1<0的值域是SKIPIF1<0值域的子集【典例1-1】．（2021上·江蘇南京·高三南京市中華中學(xué)?？计谀┒x在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時(shí)SKIPIF1<0，若對(duì)任意的SKIPIF1<0，不等式SKIPIF1<0恒成立，則實(shí)數(shù)SKIPIF1<0的最大值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例1-2】（2020·湖南永州·統(tǒng)考三模）已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0.若對(duì)任意的SKIPIF1<0，SKIPIF1<0成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1-1】（2021上·上海浦東新·高三上海市建平中學(xué)?？茧A段練習(xí)）已知SKIPIF1<0，滿足對(duì)于任意的SKIPIF1<0，都有SKIPIF1<0，設(shè)SKIPIF1<0，若對(duì)于任意的SKIPIF1<0，SKIPIF1<0，都有SKIPIF1<0成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是．【變式1-2】．（2018上·上海奉賢·高一上海市奉賢中學(xué)校考階段練習(xí)）設(shè)函數(shù)SKIPIF1<0，對(duì)任意非零實(shí)數(shù)SKIPIF1<0，若等式SKIPIF1<0成立，則正整數(shù)SKIPIF1<0的值為.【變式1-3】已知SKIPIF1<0是定義在R上的函數(shù)，且SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱.當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，若對(duì)任意的SKIPIF1<0，不等式SKIPIF1<0恒成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型14構(gòu)造“對(duì)稱”型函數(shù)【典例1-1】（2021上·湖北·高三校聯(lián)考階段練習(xí)）已知SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．前三個(gè)答案都不對(duì)【典例1-2】（2022上·上海徐匯·高三上海市南洋模范中學(xué)?？茧A段練習(xí)）設(shè)SKIPIF1<0且滿足SKIPIF1<0，則SKIPIF1<0.【變式1-1】（2022·全國(guó)·高三專題練習(xí)）已知SKIPIF1<0，那么SKIPIF1<0的值是．【變式1-2】（2021上·浙江寧波·高三余姚中學(xué)?？迹┮阎猄KIPIF1<0滿足SKIPIF1<0，若對(duì)任意的SKIPIF1<0，SKIPIF1<0恒成立，則實(shí)數(shù)k的最小值為．高考練場(chǎng)1.（2022秋·云南保山·高三統(tǒng)考階段練習(xí)）設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0是奇函數(shù)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02..已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，若函數(shù)SKIPIF1<0與SKIPIF1<0圖像的交點(diǎn)為SKIPIF1<0，則SKIPIF1<0____________.3.（2023上·貴州貴陽·高三校聯(lián)考階段練習(xí)）已知函數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0.4.（2023上·上海閔行·高三校聯(lián)考期中）設(shè)曲線SKIPIF1<0與函數(shù)SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對(duì)稱，設(shè)曲線SKIPIF1<0仍然是某函數(shù)的圖像，則實(shí)數(shù)SKIPIF1<0的取值范圍是.5.已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足：SKIPIF1<0，SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06.．（2023秋·重慶九龍