新高考數(shù)學(xué)二輪復(fù)習(xí)考點(diǎn)講義第六講基本初等函數(shù)（原卷版）

第六講：基本初等函數(shù)【考點(diǎn)梳理】1．冪函數(shù)的概念一般地，形如SKIPIF1<0(SKIPIF1<0)的函數(shù)稱為冪函數(shù)，其中底數(shù)SKIPIF1<0為自變量，SKIPIF1<0為常數(shù).2．幾個(gè)常見冪函數(shù)的圖象與性質(zhì)函數(shù)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0圖象定義域SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0值域SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0奇偶性奇函數(shù)偶函數(shù)奇函數(shù)非奇非偶函數(shù)奇函數(shù)單調(diào)性在SKIPIF1<0上單調(diào)遞增在SKIPIF1<0上單調(diào)遞減；在SKIPIF1<0上單調(diào)遞增在SKIPIF1<0上單調(diào)遞增在SKIPIF1<0上單調(diào)遞增在SKIPIF1<0和SKIPIF1<0上單調(diào)遞減過(guò)定點(diǎn)過(guò)定點(diǎn)SKIPIF1<0過(guò)定點(diǎn)SKIPIF1<03．常用結(jié)論(1)冪函數(shù)在SKIPIF1<0上都有定義.(2)冪函數(shù)的圖象均過(guò)定點(diǎn)SKIPIF1<0.(3)當(dāng)SKIPIF1<0時(shí)，冪函數(shù)的圖象均過(guò)定點(diǎn)SKIPIF1<0，且在SKIPIF1<0上單調(diào)遞增.(4)當(dāng)SKIPIF1<0時(shí)，冪函數(shù)的圖象均過(guò)定點(diǎn)SKIPIF1<0，且在SKIPIF1<0上單調(diào)遞減.(5)冪函數(shù)在第四象限無(wú)圖象.4.根式的概念及性質(zhì)（1）概念：式子SKIPIF1<0叫做根式，其中SKIPIF1<0叫做根指數(shù)，SKIPIF1<0叫做被開方數(shù)．（2）性質(zhì)：①SKIPIF1<0(SKIPIF1<0且SKIPIF1<0)；②當(dāng)SKIPIF1<0為奇數(shù)時(shí)，SKIPIF1<0；當(dāng)SKIPIF1<0為偶數(shù)時(shí)，SKIPIF1<05.分?jǐn)?shù)指數(shù)冪①正數(shù)的正分?jǐn)?shù)指數(shù)冪的意義是SKIPIF1<0(SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0)；②正數(shù)的負(fù)分?jǐn)?shù)指數(shù)冪的意義是SKIPIF1<0(SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0)；③0的正分?jǐn)?shù)指數(shù)冪等于0；0的負(fù)分?jǐn)?shù)指數(shù)冪沒(méi)有意義．6.指數(shù)冪的運(yùn)算性質(zhì)①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0.7.指數(shù)函數(shù)及其性質(zhì)（1）指數(shù)函數(shù)的概念函數(shù)SKIPIF1<0(SKIPIF1<0，且SKIPIF1<0)叫做指數(shù)函數(shù)，其中指數(shù)SKIPIF1<0是自變量，函數(shù)的定義域是SKIPIF1<0．（2）指數(shù)函數(shù)SKIPIF1<0的圖象和性質(zhì)底數(shù)SKIPIF1<0SKIPIF1<0圖象性質(zhì)定義域?yàn)镾KIPIF1<0，值域?yàn)镾KIPIF1<0圖象過(guò)定點(diǎn)SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，恒有SKIPIF1<0；當(dāng)SKIPIF1<0時(shí)，恒有SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，恒有SKIPIF1<0；當(dāng)SKIPIF1<0時(shí)，恒有SKIPIF1<0在定義域SKIPIF1<0上為增函數(shù)在定義域SKIPIF1<0上為減函數(shù)注意指數(shù)函數(shù)SKIPIF1<0(SKIPIF1<0，且SKIPIF1<0)的圖象和性質(zhì)與SKIPIF1<0的取值有關(guān)，應(yīng)分SKIPIF1<0與SKIPIF1<0來(lái)研究8.對(duì)數(shù)的概念（1）對(duì)數(shù)：一般地，如果SKIPIF1<0SKIPIF1<0，那么數(shù)SKIPIF1<0叫做以SKIPIF1<0為底SKIPIF1<0的對(duì)數(shù)，記作SKIPIF1<0，其中SKIPIF1<0叫做對(duì)數(shù)的底數(shù)，SKIPIF1<0叫做真數(shù).（2）牢記兩個(gè)重要對(duì)數(shù)：常用對(duì)數(shù)，以10為底的對(duì)數(shù)SKIPIF1<0；自然對(duì)數(shù)，以無(wú)理數(shù)e=2.71828…為底數(shù)的對(duì)數(shù)SKIPIF1<0.（3）對(duì)數(shù)式與指數(shù)式的互化：SKIPIF1<0.9.對(duì)數(shù)的性質(zhì)、運(yùn)算性質(zhì)與換底公式（1）對(duì)數(shù)的性質(zhì)根據(jù)對(duì)數(shù)的概念，知對(duì)數(shù)SKIPIF1<0具有以下性質(zhì)：①負(fù)數(shù)和零沒(méi)有對(duì)數(shù)，即SKIPIF1<0；②1的對(duì)數(shù)等于0，即SKIPIF1<0；③底數(shù)的對(duì)數(shù)等于1，即SKIPIF1<0；④對(duì)數(shù)恒等式SKIPIF1<0.（2）對(duì)數(shù)的運(yùn)算性質(zhì)如果SKIPIF1<0，那么：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0.（3）對(duì)數(shù)的換底公式對(duì)數(shù)的換底公式：SKIPIF1<0.換底公式將底數(shù)不同的對(duì)數(shù)轉(zhuǎn)化為底數(shù)相同的對(duì)數(shù)，進(jìn)而進(jìn)行化簡(jiǎn)、計(jì)算或證明.換底公式應(yīng)用時(shí)究竟換成什么為底，由已知條件來(lái)確定，一般換成以10為底的常用對(duì)數(shù)或以SKIPIF1<0為底的自然對(duì)數(shù).換底公式的變形及推廣：①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0(其中SKIPIF1<0，SKIPIF1<0，SKIPIF1<0均大于0且不等于1，SKIPIF1<0).10.對(duì)數(shù)函數(shù)及其性質(zhì)（1）對(duì)數(shù)函數(shù)的定義形如SKIPIF1<0(SKIPIF1<0，且SKIPIF1<0)的函數(shù)叫做對(duì)數(shù)函數(shù)，其中SKIPIF1<0是自變量，函數(shù)的定義域是SKIPIF1<0．（2）對(duì)數(shù)函數(shù)的圖象與性質(zhì)SKIPIF1<0SKIPIF1<0圖象性質(zhì)定義域：SKIPIF1<0值域：SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，即當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0在SKIPIF1<0上是單調(diào)增函數(shù)在SKIPIF1<0上是單調(diào)減函數(shù)【典型題型講解】考點(diǎn)一：冪函數(shù)的定義及其圖像【典例例題】例1．冪函數(shù)SKIPIF1<0在SKIPIF1<0上為增函數(shù)，則實(shí)數(shù)SKIPIF1<0的值為（

）A．SKIPIF1<0 B．0或2 C．0 D．2例2．已知冪函數(shù)SKIPIF1<0（p，q∈Z且p，q互質(zhì)）的圖象關(guān)于y軸對(duì)稱，如圖所示，則（

）A．p，q均為奇數(shù)，且SKIPIF1<0B．q為偶數(shù)，p為奇數(shù)，且SKIPIF1<0C．q為奇數(shù)，p為偶數(shù)，且SKIPIF1<0D．q為奇數(shù)，p為偶數(shù)，且SKIPIF1<0【方法技巧與總結(jié)】1.5種特殊冪函數(shù)的圖像及其性質(zhì)；2.冪函數(shù)的單調(diào)性及奇偶性的性質(zhì)判斷方法.【變式訓(xùn)練】1．（2022·廣東深圳·高三期末）已知函數(shù)SKIPIF1<0的圖像關(guān)于原點(diǎn)對(duì)稱，且在定義域內(nèi)單調(diào)遞增，則滿足上述條件的冪函數(shù)可以為SKIPIF1<0______．2.已知冪函數(shù)SKIPIF1<0（SKIPIF1<0）的圖象關(guān)于SKIPIF1<0軸對(duì)稱，且在SKIPIF1<0上是減函數(shù)，則SKIPIF1<0的值為______.3.如圖是冪函數(shù)SKIPIF1<0（αi＞0，i＝1，2，3，4，5）在第一象限內(nèi)的圖象，其中α1＝3，α2＝2，α3＝1，SKIPIF1<0，SKIPIF1<0，已知它們具有性質(zhì)：①都經(jīng)過(guò)點(diǎn)（0，0）和（1，1）；

②在第一象限都是增函數(shù)．請(qǐng)你根據(jù)圖象寫出它們?cè)冢?，+∞）上的另外一個(gè)共同性質(zhì)：___________．4.已知函數(shù)SKIPIF1<0，若關(guān)于SKIPIF1<0的方程SKIPIF1<0有兩個(gè)不同的實(shí)根，則實(shí)數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0考點(diǎn)二：指數(shù)與指數(shù)冪的運(yùn)算【典例例題】例1．化簡(jiǎn)：（1）SKIPIF1<0（2）SKIPIF1<0(a>0，b>0).（3）SKIPIF1<0.【方法技巧與總結(jié)】利用指數(shù)的運(yùn)算性質(zhì)解題.對(duì)于形如SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的形式常用“化同底”轉(zhuǎn)化，再利用指數(shù)函數(shù)單調(diào)性解決；【變式訓(xùn)練】1．SKIPIF1<0=（）A．2 B．1 C．3 D．02.甲?乙兩人解關(guān)于x的方程SKIPIF1<0，甲寫錯(cuò)了常數(shù)b，得到的根為SKIPIF1<0或x=SKIPIF1<0，乙寫錯(cuò)了常數(shù)c，得到的根為SKIPIF1<0或SKIPIF1<0，則原方程的根是（

）A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0考點(diǎn)三：指數(shù)函數(shù)的圖像及性質(zhì)【典例例題】例1.函數(shù)SKIPIF1<0恰有一個(gè)零點(diǎn)，則m的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例2.已知SKIPIF1<0為定義在R上的奇函數(shù)，SKIPIF1<0，且SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，在SKIPIF1<0上單調(diào)遞減，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【方法技巧與總結(jié)】指數(shù)函數(shù)的解析式具有單一性；指數(shù)函數(shù)的單調(diào)性和圖像與底數(shù)有關(guān)系.【變式訓(xùn)練】1.函數(shù)SKIPIF1<0，下列關(guān)于函數(shù)SKIPIF1<0的說(shuō)法錯(cuò)誤的是（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于原點(diǎn)對(duì)稱B．函數(shù)SKIPIF1<0的值域?yàn)镾KIPIF1<0C．不等式SKIPIF1<0的解集是SKIPIF1<0D．SKIPIF1<0是增函數(shù)2.函數(shù)SKIPIF1<0圖象過(guò)定點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，則SKIPIF1<0最小值為___________.3.已知定義在R上的函數(shù)SKIPIF1<0滿足：①SKIPIF1<0；②SKIPIF1<0；③在SKIPIF1<0上的解析式為SKIPIF1<0，則函數(shù)SKIPIF1<0與函數(shù)SKIPIF1<0的圖象在區(qū)間SKIPIF1<0上的交點(diǎn)個(gè)數(shù)為(

)A．3 B．4 C．5 D．64．（2022·北京·二模）若函數(shù)SKIPIF1<0的定義域和值域的交集為空集，則正數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．（2022·甘肅省武威第一中學(xué)模擬預(yù)測(cè)（文））已知函數(shù)SKIPIF1<0，則SKIPIF1<0______．6．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0______．7．已知函數(shù)SKIPIF1<0，則不等式SKIPIF1<0的解集為___________.8．設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0是函數(shù)SKIPIF1<0的最大值，則實(shí)數(shù)SKIPIF1<0的取值范圍為_______．考點(diǎn)四：對(duì)數(shù)概念與對(duì)數(shù)運(yùn)算【典例例題】例1.（1）計(jì)算SKIPIF1<0；（2）已知SKIPIF1<0，求實(shí)數(shù)x的值；（3）若SKIPIF1<0，SKIPIF1<0，用a，b，表示SKIPIF1<0．【方法技巧與總結(jié)】對(duì)數(shù)的有關(guān)運(yùn)算問(wèn)題要注意公式的順用、逆用、變形用等.對(duì)數(shù)方程或?qū)?shù)不等式問(wèn)題是要將其化為同底，利用對(duì)數(shù)單調(diào)性去掉對(duì)數(shù)符號(hào)，轉(zhuǎn)化為不含對(duì)數(shù)的問(wèn)題，但這里必須注意對(duì)數(shù)的真數(shù)為正.【變式訓(xùn)練】1.（1）求SKIPIF1<0的值.（2）已知SKIPIF1<0，SKIPIF1<0，試用SKIPIF1<0，SKIPIF1<0表示SKIPIF1<02．（2022·廣東惠州·一模）中國(guó)的5G技術(shù)領(lǐng)先世界，5G技術(shù)的數(shù)學(xué)原理之一便是著名的香農(nóng)公式：SKIPIF1<0，它表示：在受噪聲干擾的信道中，最大信息傳遞速率C取決于信道帶寬W?信道內(nèi)信號(hào)的平均功率S?信道內(nèi)部的高斯噪聲功率N的大小，其中SKIPIF1<0叫做信噪比.當(dāng)信噪比比較大時(shí)，公式中真數(shù)中的1可以忽略不計(jì)，按照香農(nóng)公式，若不改變帶寬W，而將信噪比SKIPIF1<0從1000提升至5000，則C大約增加了（

）（附：SKIPIF1<0）A．20% B．23% C．28% D．50%3．（2022·廣東韶關(guān)·一模）某種放射性物質(zhì)不斷變化為其他物質(zhì)，每經(jīng)過(guò)一年剩留的該種放射性物質(zhì)的質(zhì)量約是原來(lái)的SKIPIF1<0，估計(jì)經(jīng)過(guò)多少年，該物質(zhì)剩留的是原來(lái)的SKIPIF1<0？（

）（參考數(shù)據(jù)：SKIPIF1<0）A．16 B．17 C．18 D．194．（2022·廣東·金山中學(xué)高三期末）教室通風(fēng)的目的是通過(guò)空氣的流動(dòng)，排出室內(nèi)的污濁空氣和致病微生物，降低室內(nèi)二氧化碳和致病微生物的濃度，送進(jìn)室外的新鮮空氣.按照國(guó)家標(biāo)準(zhǔn)，教室內(nèi)空氣中二氧化碳日平均最高容許濃度應(yīng)小于等于0.1%.經(jīng)測(cè)定，剛下課時(shí)，空氣中含有0.2%的二氧化碳，若開窗通風(fēng)后教室內(nèi)二氧化碳的濃度為SKIPIF1<0%，且SKIPIF1<0隨時(shí)間SKIPIF1<0（單位：分鐘）的變化規(guī)律可以用函數(shù)SKIPIF1<0描述，則該教室內(nèi)的二氧化碳濃度達(dá)到國(guó)家標(biāo)準(zhǔn)至少需要的時(shí)間為（

）（參考數(shù)據(jù)SKIPIF1<0）A．11分鐘 B．14分鐘C．15分鐘 D．20分鐘考點(diǎn)五：對(duì)數(shù)函數(shù)的圖像及性質(zhì)【典例例題】例1．（2022·廣東中山·高三期末）已知函數(shù)SKIPIF1<0（SKIPIF1<0，SKIPIF1<0），則SKIPIF1<0的圖象可能是（

）A． B．C． D．例2．（2022·廣東珠?！じ呷谀┰O(shè)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則a，b，c大小關(guān)系為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【方法技巧與總結(jié)】對(duì)數(shù)的函數(shù)的圖像畫法，定點(diǎn)問(wèn)題；對(duì)數(shù)函數(shù)的圖像及性質(zhì)應(yīng)用.【變式訓(xùn)練】1．（2022·廣東茂名·一模）已知SKIPIF1<0均為大于0的實(shí)數(shù)，且SKIPIF1<0，則SKIPIF1<0大小關(guān)系正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·廣東茂名·一模）已知函數(shù)SKIPIF1<0，若SKIPIF1<0均不相等，且SKIPIF1<0，則SKIPIF1<0的取值范圍是___________3．（2022·廣東湛江·一模）已知函數(shù)SKIPIF1<0，SKIPIF1<0，用SKIPIF1<0表示m，n中的最小值，設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0恰有3個(gè)零點(diǎn)，則實(shí)數(shù)a的取值范圍是___________.4.己知實(shí)數(shù)SKIPIF1<0，且SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05.（多選題）已知函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）的圖象如下所示．函數(shù)SKIPIF1<0的圖象上有兩個(gè)不同的點(diǎn)SKIPIF1<0，SKIPIF1<0，則（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0在SKIPIF1<0上是奇函數(shù)C．SKIPIF1<0在SKIPIF1<0上是單調(diào)遞增函數(shù) D．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<06.（2022·廣東·三模）已知SKIPIF1<0，e是自然對(duì)數(shù)的底，若SKIPIF1<0，則SKIPIF1<0的取值可以是（

）A．1 B．2 C．3 D．4【鞏固練習(xí)】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2．1947年，生物學(xué)家MaxKleiber發(fā)表了一篇題為《bodysizeandmetabolicrate》的論文，在論文中提出了一個(gè)克萊伯定律：對(duì)于哺乳動(dòng)物，其基礎(chǔ)代謝率與體重的SKIPIF1<0次冪成正比，即SKIPIF1<0，其中F為基礎(chǔ)代謝率，M為體重．若某哺乳動(dòng)物經(jīng)過(guò)一段時(shí)間生長(zhǎng)，其體重為原來(lái)的10倍，則基礎(chǔ)代謝率為原來(lái)的（參考數(shù)據(jù)：SKIPIF1<0）（

）A．5.4倍 B．5.5倍 C．5.6倍 D．5.7倍3．已知函數(shù)SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．26 B．16 C．－16 D．－264．若函數(shù)SKIPIF1<0的零點(diǎn)為SKIPIF1<0，則SKIPIF1<0（

）．A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．25．已知函數(shù)SKIPIF1<0滿足：對(duì)任意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．關(guān)于函數(shù)SKIPIF1<0和實(shí)數(shù)SKIPIF1<0的下列結(jié)論中正確的是（

）A．若SKIPIF1<0，則SKIPIF1<0 B．若SKIPIF1<0，則SKIPIF1<0C．若SKIPIF1<0，則SKIPIF1<0 D．若SKIPIF1<0，則SKIPIF1<07．區(qū)塊鏈作為一種新型的技術(shù)，被應(yīng)用于許多領(lǐng)域.在區(qū)塊鏈技術(shù)中，某個(gè)密碼的長(zhǎng)度設(shè)定為512B，則密碼一共有SKIPIF1<0種可能，為了破解該密碼，在最壞的情況下，需要進(jìn)行SKIPIF1<0次運(yùn)算.現(xiàn)在有一臺(tái)計(jì)算機(jī)，每秒能進(jìn)行SKIPIF1<0次運(yùn)算，那么在最壞的情況下，這臺(tái)計(jì)算機(jī)破譯該密碼所需的時(shí)間大約為（參考數(shù)據(jù)SKIPIF1<0，SKIPIF1<0）（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．已知SKIPIF1<0，SKIPIF1<0，其中SKIPIF1<0且SKIPIF1<0，SKIPIF1<0且SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．39．已知正實(shí)數(shù)x，y，z滿足SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<