學(xué)生再讀教材 第二章　函數(shù)

第二章函數(shù)[知識(shí)網(wǎng)絡(luò)][命題方向]1．函數(shù)是高中數(shù)學(xué)的主干內(nèi)容，高考對(duì)本章考查的內(nèi)容主要有：(1)結(jié)合函數(shù)的定義綜合考查函數(shù)的基本性質(zhì)；(2)結(jié)合一元一次不等式、一元二次不等式、指數(shù)與對(duì)數(shù)不等式考查函數(shù)的定義域；(3)考查用待定系數(shù)法、換元法等求解函數(shù)的解析式；(4)考查指數(shù)函數(shù)、對(duì)數(shù)函數(shù)、二次函數(shù)、冪函數(shù)的圖象和性質(zhì)；(5)結(jié)合函數(shù)的圖象考查函數(shù)的性質(zhì)、函數(shù)的零點(diǎn)與方程的根；(6)結(jié)合實(shí)際問題考查函數(shù)的應(yīng)用．2．函數(shù)的基本性質(zhì)與應(yīng)用是高考的高頻考點(diǎn)，以選擇題或填空題為主，分值大約10分．3．本章內(nèi)容一般不會(huì)出現(xiàn)單一知識(shí)點(diǎn)的考題，常結(jié)合函數(shù)的單調(diào)性、奇偶性、周期性命題，或?qū)⒑瘮?shù)的性質(zhì)融入函數(shù)的圖象進(jìn)行考查，本章內(nèi)容也可能與導(dǎo)數(shù)、不等式等結(jié)合命題．探究1(人教A版必修第一冊(cè)P77)(1)設(shè)A是區(qū)間D上某些自變量的值組成的集合，而且?x1，x2∈A，當(dāng)x1＜x2時(shí)，都有f(x1)＜f(x2)，我們能說函數(shù)f(x)在區(qū)間D上單調(diào)遞增嗎？你能舉例說明嗎？(2)函數(shù)的單調(diào)性是對(duì)定義域內(nèi)某個(gè)區(qū)間而言的，你能舉出在整個(gè)定義域內(nèi)是單調(diào)遞增的函數(shù)例子嗎？你能舉出在定義域內(nèi)的某些區(qū)間上單調(diào)遞增但在另一些區(qū)間上單調(diào)遞減的函數(shù)例子嗎？______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版必修第一冊(cè)P92)探究函數(shù)y＝x＋eq\f(1,x)的圖象與性質(zhì)．在初中，我們知道y＝x是正比例函數(shù)，y＝eq\f(1,x)是反比例函數(shù)．學(xué)習(xí)了冪函數(shù)以后，我們知道它們都是冪函數(shù)．不同的函數(shù)通過加、減、乘、除等運(yùn)算可以構(gòu)成新的函數(shù)．那么，將這兩個(gè)函數(shù)相加構(gòu)成的函數(shù)有哪些性質(zhì)？這些性質(zhì)與這兩個(gè)函數(shù)的性質(zhì)有聯(lián)系嗎？下面請(qǐng)同學(xué)們帶著問題探究一下函數(shù)y＝x＋eq\f(1,x).1．你認(rèn)為可以從哪些方面研究這個(gè)函數(shù)？2．你認(rèn)為可以按照怎樣的路徑研究這個(gè)函數(shù)？______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究3(人教A版必修第一冊(cè)P135)互為反函數(shù)的兩個(gè)函數(shù)圖象間的關(guān)系．我們知道，指數(shù)函數(shù)y＝ax(a＞0，且a≠1)與對(duì)數(shù)函數(shù)y＝logax(a＞0，且a≠1)互為反函數(shù)．它們的圖象是否有關(guān)系？有什么關(guān)系呢？下面，請(qǐng)你運(yùn)用所學(xué)的數(shù)學(xué)知識(shí)和計(jì)算工具，探索幾個(gè)問題，親自發(fā)現(xiàn)其中的奧秘吧！1．在同一直角坐標(biāo)系中，畫出指數(shù)函數(shù)y＝2x及其反函數(shù)y＝log2x的圖象．你能發(fā)現(xiàn)這兩個(gè)函數(shù)的圖象有什么對(duì)稱關(guān)系嗎？_________________________________________________________________________________________________________________________________________________________________________________________________________2．取y＝2x圖象上的幾個(gè)點(diǎn)，如P1eq\b\lc\(\rc\)(\a\vs4\al\co1(－1，\f(1,2)))，P2(0，1)，P3(1，2)，P1，P2，P3關(guān)于直線y＝x的對(duì)稱點(diǎn)的坐標(biāo)是什么？它們?cè)趛＝log2x的圖象上嗎？為什么？________________________________________________________________________________________________________________________________________________________________________________________________________3．如果點(diǎn)P0(x0，y0)在函數(shù)y＝2x的圖象上，那么P0關(guān)于直線y＝x的對(duì)稱點(diǎn)在函數(shù)y＝log2x的圖象上嗎？為什么？________________________________________________________________________________________________________________________________________________________________________________________________________4．根據(jù)上述探究過程，你可以得到什么結(jié)論？___________________________________________________________________________________________________________________________________________________________________________________________________________5．上述結(jié)論對(duì)于指數(shù)函數(shù)y＝ax(a＞0，且a≠1)及其反函數(shù)y＝logax(a＞0，且a≠1)也成立嗎？為什么？________________________________________________________________________________________________________________________________________典題1(人教A版必修第一冊(cè)P66例3)下列函數(shù)中哪個(gè)與函數(shù)y＝x是同一個(gè)函數(shù)？(1)y＝(eq\r(x))2；(2)u＝eq\r(3,v3)；(3)y＝eq\r(x2)；(4)m＝eq\f(n2,n).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題2(人教A版必修第一冊(cè)P74習(xí)題3.1T13)函數(shù)f(x)＝[x]的函數(shù)值表示不超過x的最大整數(shù)，例如，[－3.5]＝－4，[2.1]＝2.當(dāng)x∈(－2.5，3]時(shí)，寫出函數(shù)f(x)的解析式，并畫出函數(shù)的圖象．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題3(人教A版必修第一冊(cè)P81例5)已知函數(shù)f(x)＝eq\f(2,x－1)(x∈[2，6])，求函數(shù)的最大值和最小值．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題4(人教A版必修第一冊(cè)P87習(xí)題3.2T12)已知函數(shù)f(x)是偶函數(shù)，而且在(0，＋∞)上單調(diào)遞減，判斷f(x)在(－∞，0)上單調(diào)遞增還是單調(diào)遞減，并證明你的判斷．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題5(人教A版必修第一冊(cè)P87習(xí)題3.2T13)我們知道，函數(shù)y＝f(x)的圖象關(guān)于坐標(biāo)原點(diǎn)成中心對(duì)稱圖形的充要條件是函數(shù)y＝f(x)為奇函數(shù)，有同學(xué)發(fā)現(xiàn)可以將其推廣為：函數(shù)y＝f(x)的圖象關(guān)于點(diǎn)P(a，b)成中心對(duì)稱圖形的充要條件是函數(shù)y＝f(x＋a)－b為奇函數(shù)．(1)求函數(shù)f(x)＝x3－3x2圖象的對(duì)稱中心；(2)類比上述推廣結(jié)論，寫出“函數(shù)y＝f(x)的圖象關(guān)于y軸成軸對(duì)稱圖形的充要條件是函數(shù)y＝f(x)為偶函數(shù)”的一個(gè)推廣結(jié)論．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·全國(guó)乙卷)設(shè)函數(shù)f(x)＝eq\f(1－x,1＋x)，則下列函數(shù)中為奇函數(shù)的是()A．f(x－1)－1 B．f(x－1)＋1C．f(x＋1)－1 D．f(x＋1)＋1點(diǎn)評(píng)本題是教材習(xí)題結(jié)論的直接應(yīng)用，如果利用結(jié)論解答該題更簡(jiǎn)單，解法如下：f(x)＝eq\f(1－x,1＋x)＝－1＋eq\f(2,1＋x)，則f(x)的圖象關(guān)于點(diǎn)P(－1，－1)對(duì)稱，故函數(shù)y＝f(x－1)＋1為奇函數(shù)．典題6(人教A版必修第一冊(cè)P96習(xí)題3.4T4)圖(1)是某條公共汽車線路收支差額y關(guān)于乘客量x的圖象．(1)試說明圖(1)上點(diǎn)A，點(diǎn)B以及射線AB上的點(diǎn)的實(shí)際意義；(2)由于目前本條線路虧損，公司有關(guān)人員提出了兩種扭虧為贏的建議，如圖(2)(3)所示．你能根據(jù)圖象，說明這兩種建議是什么嗎？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題7(人教A版必修第一冊(cè)P65例2)已知函數(shù)f(x)＝eq\r(x＋3)＋eq\f(1,x＋2)，當(dāng)a＞0時(shí)，求f(a)，f(a－1)的值．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·新高考Ⅱ卷)已知函數(shù)f(x)的定義域?yàn)镽，且f(x＋y)＋f(x－y)＝f(x)f(y)，f(1)＝1，則eq\o(∑,\s\up6(22),\s\do4(k＝1))f(k)＝()A．－3 B．－2C．0 D．1點(diǎn)評(píng)本題與教材例題都考查的是函數(shù)的賦值與整體代換的思想．典題8(人教A版必修第一冊(cè)P101復(fù)習(xí)參考題3T9)(1)已知奇函數(shù)f(x)在[a，b]上單調(diào)遞減，那么它在[－b，－a]上單調(diào)遞增還是單調(diào)遞減？(2)已知偶函數(shù)g(x)在[a，b]上單調(diào)遞減，那么它在[－b，－a]上單調(diào)遞增還是單調(diào)遞減？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題9(人教A版必修第一冊(cè)P101復(fù)習(xí)參考題3T12)試討論函數(shù)y＝x－eq\f(1,x)的定義域、值域、單調(diào)性、奇偶性，并畫出函數(shù)圖象．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版必修第一冊(cè)P117例3)比較下列各題中兩個(gè)值的大小：(1)1.72.5，1.73；(2)0.8－eq\r(2)，0.8－eq\r(3)；(3)1.70.3，0.93.1._________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·天津卷)若a＝1.010.5，b＝1.010.6，c＝0.60.5，則a，b，c的大小關(guān)系為()A．c>a>b B．c>b>aC．a(chǎn)>b>c D．b>a>c點(diǎn)評(píng)教材例題體現(xiàn)了比較大小的兩種常用方法：(1)利用函數(shù)的單調(diào)性；(2)借助于0或1作為中間數(shù)，而高考試題考查的也正是這兩種方法．典題11(人教A版必修第一冊(cè)P120習(xí)題4.2T9)已知函數(shù)y＝aeq\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,2)))eq\s\up12(|x|)＋b的圖象過原點(diǎn)，且無限接近直線y＝2但又不與該直線相交．(1)求該函數(shù)的解析式，并畫出圖象；(2)判斷該函數(shù)的奇偶性和單調(diào)性．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題12(人教A版必修第一冊(cè)P126例5)盡管目前人類還無法準(zhǔn)確預(yù)報(bào)地震，但科學(xué)家通過研究，已經(jīng)對(duì)地震有所了解，例如，地震時(shí)釋放出的能量E(單位：焦耳)與地震里氏震級(jí)M之間的關(guān)系為lgE＝4.8＋1.5M.2011年3月11日，日本東北部海域發(fā)生里氏9.0級(jí)地震，它所釋放出來的能量是2008年5月12日我國(guó)汶川發(fā)生里氏8.0級(jí)地震的多少倍(精確到1)?_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·全國(guó)甲卷)青少年視力是社會(huì)普遍關(guān)注的問題，視力情況可借助視力表測(cè)量，通常用五分記錄法和小數(shù)記錄法記錄視力數(shù)據(jù)，五分記錄法的數(shù)據(jù)L和小數(shù)記錄法的數(shù)據(jù)V滿足L＝5＋lgV．已知某同學(xué)視力的五分記錄法的數(shù)據(jù)為4.9，則其視力的小數(shù)記錄法的數(shù)據(jù)約為(eq\r(10,10)≈1.259)()A．1.5 B．1.2C．0.8 D．0.6點(diǎn)評(píng)本題與教材例題類似，主要考查兩個(gè)方面知識(shí)：一是指數(shù)、對(duì)數(shù)的互化與運(yùn)算；二是利用數(shù)學(xué)知識(shí)解決實(shí)際問題．典題13(人教A版必修第一冊(cè)P133例3)比較下列各題中兩個(gè)值的大小：(1)log23.4，log28.5；(2)log0.31.8，log0.32.7；(3)loga5.1，loga5.9(a＞0，且a≠1)．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2020·全國(guó)Ⅲ卷)設(shè)a＝log32，b＝log53，c＝eq\f(2,3)，則()A．a(chǎn)<c<b B．a(chǎn)<b<cC．b<c<a D．c<a<b點(diǎn)評(píng)本題是教材習(xí)題的拓展，由于a和b的底數(shù)不同，故不能直接利用單調(diào)性比較大小，需變形后比較大小，而變形的過程中應(yīng)用了函數(shù)的單調(diào)性．典題14(人教A版必修第一冊(cè)P140習(xí)題4.4T4)函數(shù)y＝log2x，y＝log5x，y＝lgx的圖象如圖所示．(1)試說明哪個(gè)函數(shù)對(duì)應(yīng)于哪個(gè)圖象，并解釋為什么；(2)以已有圖象為基礎(chǔ)，在同一直角坐標(biāo)系中畫出y＝logeq\s\do9(\f(1,2))x，y＝logeq\s\do9(\f(1,5))x，y＝logeq\s\do9(\f(1,10))x的圖象；(3)從(2)的圖中你發(fā)現(xiàn)了什么？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題15(人教A版必修第一冊(cè)P156習(xí)題4.5T13)有一道題“若函數(shù)f(x)＝24ax2＋4x－1在區(qū)間(－1，1)內(nèi)恰有一個(gè)零點(diǎn)，求實(shí)數(shù)a的取值范圍”，某同學(xué)給出了如下解答：由f(－1)f(1)＝(24a－5)(24a＋3)＜0，解得－eq\f(1,8)＜a＜eq\f(5,24).所以，實(shí)數(shù)a的取值范圍是eq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(1,8)，\f(5,24))).上述解答正確嗎？若不正確，請(qǐng)說明理由，并給出正確的解答．___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題16(人教A版必修第一冊(cè)P161復(fù)習(xí)參考題4T11)已知函數(shù)f(x)＝loga(x＋1)，g(x)＝loga(1－x)(a＞0，且a≠1)．(1)求函數(shù)f(x)＋g(x)的定義域；(2)判斷函數(shù)f(x)＋g(x)的奇偶性，并說明理由．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題17(人教A版必修第一冊(cè)P161復(fù)習(xí)參考題4T12)對(duì)于函數(shù)f(x)＝a－eq\f(2,2x＋1)(a∈R)．(1)探索函數(shù)f(x)的單調(diào)性；(2)是否存在實(shí)數(shù)a使函數(shù)f(x)為奇函數(shù)？____________________________________________________________________________________________________________________