人教A版高中數(shù)學(xué)（必修第一冊(cè)）同步講義 3.1.1函數(shù)的概念及其表示（原卷版）

第第頁(yè)3.1.1函數(shù)的概念課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①函數(shù)的概念；②了解函數(shù)的三要素；③掌握簡(jiǎn)單函數(shù)的定義域；④掌握求函數(shù)的值；⑤掌握區(qū)間的寫(xiě)法.通過(guò)本節(jié)課的學(xué)習(xí)，掌握函數(shù)概念及函數(shù)的三要素，會(huì)判斷同一函數(shù)，會(huì)求簡(jiǎn)單函數(shù)的定義域及值域.知識(shí)點(diǎn)01：函數(shù)的概念1、初中學(xué)習(xí)的函數(shù)的傳統(tǒng)定義設(shè)在一個(gè)變化的過(guò)程中，有兩個(gè)變量SKIPIF1<0和SKIPIF1<0，如果給定了一個(gè)SKIPIF1<0值，相應(yīng)地就有唯一確定的一個(gè)SKIPIF1<0值與之對(duì)應(yīng)，那么我們就稱SKIPIF1<0是SKIPIF1<0的函數(shù)，其中SKIPIF1<0是自變量，SKIPIF1<0是因變量.它們描述的是兩個(gè)變量之間的依賴關(guān)系.2、函數(shù)的近代定義一般地，設(shè)SKIPIF1<0，SKIPIF1<0是非空的實(shí)數(shù)集，如果對(duì)于集合SKIPIF1<0中的任意一個(gè)數(shù)SKIPIF1<0，按照某種確定的對(duì)應(yīng)關(guān)系SKIPIF1<0，在集合SKIPIF1<0中都有唯一確定的數(shù)SKIPIF1<0和它對(duì)應(yīng)，那么就稱SKIPIF1<0為從集合SKIPIF1<0到集合SKIPIF1<0的一個(gè)函數(shù)(function)，記作SKIPIF1<0，SKIPIF1<0.其中，SKIPIF1<0叫做自變量，SKIPIF1<0的取值范圍SKIPIF1<0叫做函數(shù)的定義域；與SKIPIF1<0的值相對(duì)應(yīng)的SKIPIF1<0值叫做函數(shù)值，函數(shù)值的集合SKIPIF1<0叫做函數(shù)的值域．顯然，值域是集合SKIPIF1<0的子集.函數(shù)的四個(gè)特征：①非空性：SKIPIF1<0，SKIPIF1<0必須為非空數(shù)集（注意不僅非空，還要是數(shù)集），定義域或值域?yàn)榭占暮瘮?shù)是不存在的.②任意性：即定義域中的每一個(gè)元素都有函數(shù)值.③單值性：每一個(gè)自變量有且僅有唯一的函數(shù)值與之對(duì)應(yīng)（可以多對(duì)一，不能一對(duì)多）.④方向性：函數(shù)是一個(gè)從定義域到值域的對(duì)應(yīng)關(guān)系，如果改變這個(gè)對(duì)應(yīng)方向，那么新的對(duì)應(yīng)所確定的關(guān)系就不一定是函數(shù)關(guān)系.【即學(xué)即練1】（多選）下列四個(gè)圖象中，是函數(shù)圖象的是（

）A．

B．

C．

D．

知識(shí)點(diǎn)02：函數(shù)的三要素1、定義域：函數(shù)的定義域是自變量的取值范圍．2、對(duì)應(yīng)關(guān)系：對(duì)應(yīng)關(guān)系SKIPIF1<0是函數(shù)的核心，它是對(duì)自變量SKIPIF1<0實(shí)施“對(duì)應(yīng)操作”的“程序”或者“方法”．3、值域：與SKIPIF1<0的值相對(duì)應(yīng)的SKIPIF1<0值叫做函數(shù)值，函數(shù)值的集合SKIPIF1<0叫做函數(shù)的值域(range).【即學(xué)即練2】函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_____．知識(shí)點(diǎn)03：函數(shù)相等同一函數(shù)：只有當(dāng)兩個(gè)函數(shù)的定義域和對(duì)應(yīng)關(guān)系都分別相同時(shí),這兩個(gè)函數(shù)才相等,即是同一個(gè)函數(shù).【即學(xué)即練3】下列四組函數(shù)中，表示同一函數(shù)的是（

）A．SKIPIF1<0與SKIPIF1<0B．SKIPIF1<0與SKIPIF1<0C．SKIPIF1<0與SKIPIF1<0D．SKIPIF1<0與SKIPIF1<0知識(shí)點(diǎn)04：區(qū)間的概念1區(qū)間的概念設(shè)SKIPIF1<0，SKIPIF1<0是實(shí)數(shù)，且SKIPIF1<0，滿足SKIPIF1<0的實(shí)數(shù)SKIPIF1<0的全體，叫做閉區(qū)間，記作SKIPIF1<0，即，SKIPIF1<0。如圖：SKIPIF1<0，SKIPIF1<0叫做區(qū)間的端點(diǎn)．在數(shù)軸上表示一個(gè)區(qū)間時(shí)，若區(qū)間包括端點(diǎn)，則端點(diǎn)用實(shí)心點(diǎn)表示；若區(qū)間不包括端點(diǎn)，則端點(diǎn)用空心點(diǎn)表示．集合SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0區(qū)間SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<02含有無(wú)窮大的表示全體實(shí)數(shù)也可用區(qū)間表示為SKIPIF1<0，符號(hào)“SKIPIF1<0”讀作“正無(wú)窮大”，“SKIPIF1<0”讀作“負(fù)無(wú)窮大”，即SKIPIF1<0。集合SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0區(qū)間SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0【即學(xué)即練4】已知集合SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型01函數(shù)關(guān)系的判斷【典例1】若函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，值域?yàn)镾KIPIF1<0，則SKIPIF1<0的圖象可能是（

）A． B．C． D．【典例2】已知集合SKIPIF1<0，SKIPIF1<0，下列對(duì)應(yīng)關(guān)系中，從SKIPIF1<0到SKIPIF1<0的函數(shù)為（

）A．f：SKIPIF1<0 B．f：SKIPIF1<0C．f：SKIPIF1<0 D．f：SKIPIF1<0【變式1】（多選）下列對(duì)應(yīng)中是函數(shù)的是（

）．A．SKIPIF1<0，其中SKIPIF1<0，SKIPIF1<0，SKIPIF1<0B．SKIPIF1<0，其中SKIPIF1<0，SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，其中y為不大于x的最大整數(shù)，SKIPIF1<0，SKIPIF1<0D．SKIPIF1<0，其中SKIPIF1<0，SKIPIF1<0，SKIPIF1<0題型02集合與區(qū)間的轉(zhuǎn)化【典例1】已知全集SKIPIF1<0,集合SKIPIF1<0,則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】若集合SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】全集SKIPIF1<0，集合SKIPIF1<0，集合SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型03同一個(gè)函數(shù)【典例1】下列各組函數(shù)表示相同函數(shù)的是（

）A．SKIPIF1<0和SKIPIF1<0 B．SKIPIF1<0和SKIPIF1<0C．SKIPIF1<0和SKIPIF1<0 D．SKIPIF1<0和SKIPIF1<0【典例2】（多選）下面各組函數(shù)表示同一函數(shù)的是（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0（SKIPIF1<0），SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0【變式1】下列每組中的函數(shù)是同一個(gè)函數(shù)的是（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0題型04求函數(shù)值【典例1】若函數(shù)SKIPIF1<0，則SKIPIF1<0_________．【典例2】若SKIPIF1<0，則SKIPIF1<0＝______．【變式1】設(shè)函數(shù)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】已知SKIPIF1<0，SKIPIF1<0．（1）計(jì)算：SKIPIF1<0____________；（2）計(jì)算：SKIPIF1<0____________．題型05根據(jù)函數(shù)值請(qǐng)求自變量或參數(shù)【典例1】若函數(shù)SKIPIF1<0的值域是SKIPIF1<0，則此函數(shù)的定義域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（多選）若函數(shù)SKIPIF1<0在定義域SKIPIF1<0上的值域?yàn)镾KIPIF1<0，則區(qū)間SKIPIF1<0可能為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】已知函數(shù)SKIPIF1<0的值域是SKIPIF1<0，則x的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型06函數(shù)的定義域（具體函數(shù)的定義域）【典例1】已知函數(shù)SKIPIF1<0的定義域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_____．【變式1】函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_______.題型07函數(shù)的定義域（抽象函數(shù)的定義域）【典例1】已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，則函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_____.【典例2】若SKIPIF1<0的定義域?yàn)镾KIPIF1<0，求SKIPIF1<0的定義域．【變式1】（1）已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，則函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_____．（2）已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，則函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_____．題型08函數(shù)的定義域（復(fù)合函數(shù)的定義域）【典例1】若函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，則函數(shù)SKIPIF1<0的定義域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0）的定義域?yàn)镾KIPIF1<0，則函數(shù)SKIPIF1<0的定義域?yàn)椋ǎ〢．（SKIPIF1<0，4）B．[SKIPIF1<0，4）C．（SKIPIF1<0，6）D．（SKIPIF1<0，2）【變式1】已知函數(shù)SKIPIF1<0，SKIPIF1<0，則函數(shù)SKIPIF1<0的定義域?yàn)開(kāi)_____.題型09函數(shù)的定義域（實(shí)際問(wèn)題中的定義域）【典例1】已知等腰三角形的周長(zhǎng)為SKIPIF1<0，底邊長(zhǎng)SKIPIF1<0是腰長(zhǎng)SKIPIF1<0的函數(shù)，則函數(shù)的定義域?yàn)?)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】周長(zhǎng)為定值SKIPIF1<0的矩形，它的面積SKIPIF1<0是這個(gè)矩形的一邊長(zhǎng)SKIPIF1<0的函數(shù)，則這個(gè)函數(shù)的定義域是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】如圖，某小區(qū)有一塊底邊和高均為40m的銳角三角形空地，現(xiàn)規(guī)劃在空地內(nèi)種植一邊長(zhǎng)為SKIPIF1<0（單位：m）的矩形草坪（陰影部分），要求草坪面積不小于SKIPIF1<0，則SKIPIF1<0的取值范圍為_(kāi)_____.題型10函數(shù)的值域（常見(jiàn)（一次，二次，反比例）函數(shù)的值域）【典例1】函數(shù)SKIPIF1<0，則SKIPIF1<0的值域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】求下列函數(shù)的定義域、值域，并畫(huà)出圖象：(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0；(4)SKIPIF1<0；(5)SKIPIF1<0；(6)SKIPIF1<0．【變式1】例題3．求下列函數(shù)的值域.(1)SKIPIF1<0;(2)SKIPIF1<0,SKIPIF1<0.題型11函數(shù)的值域（根式型函數(shù)的值域）【典例1】函數(shù)SKIPIF1<0的值域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】求函數(shù)SKIPIF1<0的值域?yàn)開(kāi)________.【變式1】函數(shù)SKIPIF1<0的值域是___________.題型12函數(shù)的值域（分式型函數(shù)的值域）【典例1】函數(shù)ySKIPIF1<0的值域是（）A．（﹣∞，+∞） B．（﹣∞，SKIPIF1<0）∪（SKIPIF1<0，+∞）C．（﹣∞，SKIPIF1<0）∪（SKIPIF1<0，+∞） D．（﹣∞，SKIPIF1<0）∪（SKIPIF1<0，+∞）【典例2】（1）求函數(shù)SKIPIF1<0的值域；（2）求函數(shù)SKIPIF1<0的值域.【變式1】求函數(shù)SKIPIF1<0的值域______________．【變式2】函數(shù)SKIPIF1<0的值域是___________.題型13根據(jù)函數(shù)的值域求定義域【典例1】已知函數(shù)SKIPIF1<0的值域是SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（多選）已知函數(shù)SKIPIF1<0的值域是SKIPIF1<0，則其定義域可能是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】（多選）若函數(shù)SKIPIF1<0在定義域SKIPIF1<0上的值域?yàn)镾KIPIF1<0，則區(qū)間SKIPIF1<0可能為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型14重點(diǎn)方法之換元法求值域【典例1】函數(shù)SKIPIF1<0的值域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】求函數(shù)SKIPIF1<0的值域.題型15重點(diǎn)方法之分離常數(shù)法求值域【典例1】求函數(shù)SKIPIF1<0的值域.【典例2】求下列函數(shù)的值域：SKIPIF1<0題型16數(shù)學(xué)思想方法（數(shù)形結(jié)合的思想方法）【典例1】已知函數(shù)SKIPIF1<0的值域是SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】若函數(shù)SKIPIF1<0的定義域是SKIPIF1<0，則其值域?yàn)椋?/p>

）.A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0題型17易錯(cuò)題（換元必?fù)Q范圍）【典例1】求下列函數(shù)的值域：SKIPIF1<0．【典例2】函數(shù)SKIPIF1<0的值域?yàn)開(kāi)__________．3.1.1函數(shù)的概念A(yù)夯實(shí)基礎(chǔ)一、單選題1．已知函數(shù)SKIPIF1<0，那么SKIPIF1<0的值（

）A．3 B．5 C．72．函數(shù)SKIPIF1<0的定義域是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<03．下列各函數(shù)中，與函數(shù)SKIPIF1<0表示同一函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．已知高斯取整函數(shù)SKIPIF1<0，則SKIPIF1<0的值為（

）A．2 B．3 C．4 D．55．下表給出了x與SKIPIF1<0和SKIPIF1<0的對(duì)應(yīng)關(guān)系，根據(jù)表格可知SKIPIF1<0的值為（

）x1234x1234SKIPIF1<03142SKIPIF1<04321A．1 B．2 C．3 D．46．下列對(duì)應(yīng)是從集合A到集合B的函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．若函數(shù)SKIPIF1<0的定義域是SKIPIF1<0，則函數(shù)SKIPIF1<0的定義域是（

）A．[-4，1] B．[-3，1] C．[-3，1） D．[-4，1）8．函數(shù)SKIPIF1<0的值域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．若函數(shù)SKIPIF1<0定義域?yàn)镾KIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則下列結(jié)果正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．中國(guó)清朝數(shù)學(xué)家李善蘭在1859年翻譯《代數(shù)學(xué)》中首次將“function”譯做：“函數(shù)”，沿用至今，為什么這么翻譯，書(shū)中解釋說(shuō)“凡此變數(shù)中函彼變數(shù)者，則此為彼之函數(shù)”．1930年美國(guó)人給出了我們課本中所學(xué)的集合論的函數(shù)定義．已知集合M＝{SKIPIF1<01，1，2，4}，N＝{1，2，4，16}，給出下列四個(gè)對(duì)應(yīng)法則，請(qǐng)由函數(shù)定義判斷，其中能構(gòu)成從M到N的函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0三、填空題11．設(shè)二次函數(shù)SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的值域是SKIPIF1<0，則SKIPIF1<0的最小值是____________．12．函數(shù)SKIPIF1<0的值域?yàn)開(kāi)__________．四、解答題13．已知函數(shù)SKIPIF1<0的定義域?yàn)锳，集合SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0；(2)若SKIPIF1<0，求a的取值范圍.14．已知函數(shù)SKIPIF1<0的值域?yàn)閇1，3]，求SKIPIF1<0的值B能力提升1．若函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，值域?yàn)镾KIPIF1<0，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．函數(shù)SKIPIF1<0的值域?yàn)锳．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．設(shè)SKIPIF1<0，用SKIPIF1<0表示不超過(guò)SKIPIF1<0的最大整數(shù)，則SKIPIF1<0稱為高斯函數(shù)．例如：SKIPIF1<0，SKIPIF1<0，已知函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的值域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知函數(shù)SKIPIF1<0．(1)求SKIPIF1<0，SKIPIF1<0的值；(2)求證：SKIPIF1<0的定值；(3)求SKIPIF1<0的值．C綜合素養(yǎng)1．歐拉函數(shù)SKIPIF1<0的函數(shù)值等于所有不超過(guò)正整數(shù)SKIPIF1<0，且與SKIPIF1<0互素的正整數(shù)的個(gè)數(shù)，例如，SKIPIF1<0，SKIPIF1<0．若SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（多選）對(duì)于定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0，若同時(shí)滿足下列條件：①SKIPIF1<0，SKIPIF1<0；②SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則稱函數(shù)SKIPIF1<0為“SKIPIF1<0函數(shù)”.下列結(jié)論正確的是（

）A．若SKIPIF1<0為“SKIPIF1<0函數(shù)”，則其圖象恒過(guò)定點(diǎn)SKIPIF1<0B．函數(shù)SKIPIF1<0在SKIPIF1<0上是“SKIPIF1<0函數(shù)”C．函數(shù)SKIPIF1<0在SKIPIF1<0上是“SKIPIF1<0函數(shù)”（SKIPIF1<0表示不大于SKIPIF1<0的最大整數(shù)）D．若SKIPIF1<0為“SKIPIF1<0函數(shù)”，則SKIPIF1<0一定是SKIPIF1<0上的增函數(shù)3．已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0，對(duì)于任意的SKIPIF1<0恒有SKIPIF1<0.(1)若SKIPIF1<0，求SKIPIF1<0的值；(2)若SKIPIF1<0，求SKIPIF1<0的值.4．如果一個(gè)函數(shù)的值域與其定義域相同，則稱該函數(shù)為“同域函數(shù)”.已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0且SKIPIF1<0.（Ⅰ）若SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的定義域；（Ⅱ）當(dāng)SKIPIF1<0時(shí)，若SKIPIF1<0為“同域函數(shù)”，求實(shí)數(shù)SKIPIF1<0的值；（Ⅲ）若存在實(shí)數(shù)SKIPIF1<0且SKIPIF1<0，使得SKIPIF1<0為“同域函數(shù)”，求實(shí)數(shù)SKIPIF1<0的取值范圍.

3.1.2函數(shù)的表示法課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①了解函數(shù)的三種表示方法及特點(diǎn)；②掌握求函數(shù)解析式的常用方法③了解與認(rèn)識(shí)分段函數(shù)及其定義域④會(huì)用分析法與圖象法表示分段函數(shù)，并能掌握分段函數(shù)的相關(guān)性質(zhì).通過(guò)本節(jié)課的學(xué)習(xí)，熟練掌握函數(shù)的三種表示方法，會(huì)求函數(shù)的解析式，掌握分段函數(shù)的解析法與圖象法的表示方法與性質(zhì).知識(shí)點(diǎn)01：函數(shù)的表示法1、解析法：用數(shù)學(xué)表達(dá)式表示兩個(gè)變量之間的對(duì)應(yīng)關(guān)系.2、列表法：列出表格來(lái)表示兩個(gè)變量之間的對(duì)應(yīng)關(guān)系.3、圖象法：用圖象表示兩個(gè)變量之間的對(duì)應(yīng)關(guān)系.優(yōu)點(diǎn)缺點(diǎn)聯(lián)系解析法①簡(jiǎn)明、全面的概括了變量之間的關(guān)系；②可以通過(guò)解析式求出在定義域內(nèi)任意自變量所對(duì)應(yīng)的函數(shù)值；③便于利用解析式研究函數(shù)的性質(zhì)；①并不是所有的函數(shù)都有解析式；②不能直觀地觀察到函數(shù)的變化規(guī)律；解析法、圖象法、列表法各有各的優(yōu)缺點(diǎn)，面對(duì)實(shí)際情境時(shí)，我們要根據(jù)不同的需要選擇恰當(dāng)?shù)姆椒ū硎竞瘮?shù)．圖象法①能直觀、形象地表示自變量的變化情況及相適應(yīng)的函數(shù)值的變化趨勢(shì)；②可以直接應(yīng)用圖象來(lái)研究函數(shù)的性質(zhì)；①并不是所有的函數(shù)都能畫(huà)出圖象；②不能精確地求出某一自變量相應(yīng)的函數(shù)值；列表法①不需要計(jì)算就可以直接看出與自變量的值對(duì)應(yīng)的函數(shù)值；①不夠全面，只能表示自變量取較少的有限值的對(duì)應(yīng)關(guān)系；②不能明顯地展示出因變量隨自變量變化的規(guī)律；知識(shí)點(diǎn)02：求函數(shù)解析式1、待定系數(shù)法：若已知函數(shù)的類型（如一次函數(shù)、二次函數(shù)，反比例等），可用待定系數(shù)法．2、換元法：主要用于解決已知SKIPIF1<0這類復(fù)合函數(shù)的解析式，求函數(shù)SKIPIF1<0的解析式的問(wèn)題，在使用換元法時(shí)特別注意，換元必?fù)Q范圍.3、配湊法：由已知條件SKIPIF1<0，可將SKIPIF1<0改寫(xiě)成關(guān)于SKIPIF1<0的表達(dá)式，4、方程組（消去）法：主要解決已知SKIPIF1<0與SKIPIF1<0、SKIPIF1<0、SKIPIF1<0……的方程，求SKIPIF1<0解析式。【即學(xué)即練1】已知SKIPIF1<0，則函數(shù)SKIPIF1<0_______，SKIPIF1<0=_______．知識(shí)點(diǎn)03：分段函數(shù)對(duì)于函數(shù)SKIPIF1<0，若自變量在定義域內(nèi)的在不同范圍取值時(shí)，函數(shù)的對(duì)應(yīng)關(guān)系也不相同，則稱函數(shù)SKIPIF1<0叫分段函數(shù)．注：(1)分段函數(shù)是一個(gè)函數(shù)，只是自變量在不同范圍取值時(shí)，函數(shù)的對(duì)應(yīng)關(guān)系不相同；（2）在書(shū)寫(xiě)時(shí)要指明各段函數(shù)自變量的取值范圍；（3）分段函數(shù)的定義域是所以自變量取值區(qū)間的并集．【即學(xué)即練2】已知函數(shù)SKIPIF1<0，則SKIPIF1<0_________．知識(shí)點(diǎn)04：函數(shù)的圖象1、函數(shù)圖象的平移變換（左“+”右“-”；上“+”下“-”）①SKIPIF1<0②SKIPIF1<0③SKIPIF1<0④SKIPIF1<0注：左右平移只能單獨(dú)一個(gè)SKIPIF1<0加或者減，注意當(dāng)SKIPIF1<0前系數(shù)不為1,需將系數(shù)提取到外面.2、函數(shù)圖象的對(duì)稱變換①SKIPIF1<0的圖象SKIPIF1<0SKIPIF1<0的圖象；②SKIPIF1<0的圖象SKIPIF1<0SKIPIF1<0的圖象；③SKIPIF1<0的圖象SKIPIF1<0SKIPIF1<0的圖象；3、函數(shù)圖象的翻折變換（絕對(duì)值變換）①SKIPIF1<0的圖象SKIPIF1<0SKIPIF1<0的圖象；（口訣；以SKIPIF1<0軸為界，保留SKIPIF1<0軸上方的圖象；將SKIPIF1<0軸下方的圖象翻折到SKIPIF1<0軸上方）②SKIPIF1<0的圖象SKIPIF1<0SKIPIF1<0的圖象.（口訣；以SKIPIF1<0軸為界，去掉SKIPIF1<0軸左側(cè)的圖象，保留SKIPIF1<0軸右側(cè)的圖象；將SKIPIF1<0軸右側(cè)圖象翻折到SKIPIF1<0軸左側(cè)；本質(zhì)是個(gè)偶函數(shù)）【即學(xué)即練3】函數(shù)SKIPIF1<0的部分圖象大致是（

）A．B．C．D．題型01函數(shù)的三種表示法的應(yīng)用【典例1】已知邊長(zhǎng)為1的正方形SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，動(dòng)點(diǎn)SKIPIF1<0在正方形SKIPIF1<0邊上沿SKIPIF1<0運(yùn)動(dòng)．設(shè)點(diǎn)SKIPIF1<0經(jīng)過(guò)的路程為SKIPIF1<0．SKIPIF1<0的面積為SKIPIF1<0．則SKIPIF1<0與SKIPIF1<0的函數(shù)圖象大致為圖中的（）A．

B．

C．

D．

【典例2】如圖中的圖象所表示的函數(shù)的解析式為（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例3】已知函數(shù)SKIPIF1<0，SKIPIF1<0分別由下表給出，SKIPIF1<0012SKIPIF1<0121SKIPIF1<0012SKIPIF1<0210則SKIPIF1<0_____________；滿足SKIPIF1<0的SKIPIF1<0的值是_____________．【變式1】如圖，SKIPIF1<0是邊長(zhǎng)為2的等邊三角形，點(diǎn)SKIPIF1<0由點(diǎn)SKIPIF1<0沿線段SKIPIF1<0向點(diǎn)SKIPIF1<0移動(dòng)，過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0的垂線SKIPIF1<0，設(shè)SKIPIF1<0，記位于直線SKIPIF1<0左側(cè)的圖形的面積為SKIPIF1<0，那么SKIPIF1<0與SKIPIF1<0的函數(shù)關(guān)系的圖象大致是（

）A．B．C．D．【變式2】某校要召開(kāi)學(xué)生代表大會(huì)，規(guī)定各班每SKIPIF1<0人推選一名代表，當(dāng)班人數(shù)除以SKIPIF1<0的余數(shù)大于SKIPIF1<0時(shí)，再增選一名代表，則各班推選代表人數(shù)SKIPIF1<0與該班人數(shù)SKIPIF1<0之間的函數(shù)關(guān)系用取整函數(shù)SKIPIF1<0（SKIPIF1<0表示不大于SKIPIF1<0的最大整數(shù)，如SKIPIF1<0，SKIPIF1<0）可表示為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型02求函數(shù)的解析式（待定系數(shù)法）【典例1】（多選）已知函數(shù)SKIPIF1<0是一次函數(shù)，滿足SKIPIF1<0，則SKIPIF1<0的解析式可能為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】若二次函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的表達(dá)式為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式1】若SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．3 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型03求函數(shù)的解析式（換元法）【典例1】已知SKIPIF1<0，則函數(shù)SKIPIF1<0的解析式為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知SKIPIF1<0，求SKIPIF1<0.題型04求函數(shù)的解析式（配湊法）【典例1】已知SKIPIF1<0，則SKIPIF1<0_______．【典例2】已知SKIPIF1<0，則SKIPIF1<0（

）A．6 B．3 C．11 D．10題型05求函數(shù)的解析式（方程組（消去）法）【典例1】已知函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0，對(duì)任意SKIPIF1<0均滿足：SKIPIF1<0則函數(shù)SKIPIF1<0解析式為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知SKIPIF1<0，求SKIPIF1<0的解析式【變式1】已知函數(shù)SKIPIF1<0滿足SKIPIF1<0．(1)求SKIPIF1<0的解析式；(2)設(shè)函數(shù)SKIPIF1<0，若對(duì)任意SKIPIF1<0，SKIPIF1<0恒成立，求實(shí)數(shù)m的取值范圍．題型06求函數(shù)的解析式（賦值法求抽象函數(shù)的解析式）【典例1】定義在SKIPIF1<0上的函數(shù)SKIPIF1<0滿足SKIPIF1<0，并且對(duì)任意實(shí)數(shù)SKIPIF1<0，SKIPIF1<0都有SKIPIF1<0，求SKIPIF1<0的解析式.【典例2】已知函數(shù)SKIPIF1<0滿足：對(duì)一切實(shí)數(shù)SKIPIF1<0、SKIPIF1<0，均有SKIPIF1<0成立，且SKIPIF1<0．求函數(shù)SKIPIF1<0的表達(dá)式；【變式1】寫(xiě)出一個(gè)滿足：SKIPIF1<0的函數(shù)解析式為_(kāi)_____.【變式2】已知函數(shù)SKIPIF1<0對(duì)一切的實(shí)數(shù)SKIPIF1<0，SKIPIF1<0，都滿足SKIPIF1<0，且SKIPIF1<0.（1）求SKIPIF1<0的值；（2）求SKIPIF1<0的解析式；（3）求SKIPIF1<0在SKIPIF1<0上的值域.題型07分段函數(shù)（求分段函數(shù)的值）【典例1】已知函數(shù)SKIPIF1<0，則SKIPIF1<0______．【典例2】已知函數(shù)SKIPIF1<0，則SKIPIF1<0的值為_(kāi)_____.題型08分段函數(shù)（已知分段函數(shù)的值求參數(shù)）【典例1】已知函數(shù)SKIPIF1<0，若SKIPIF1<0的最小值為1，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0無(wú)最大值，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【典例3】已知函數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0的最大值是________．題型09分段函數(shù)（分段函數(shù)的值域或最值）【典例1】函數(shù)SKIPIF1<0的最小值是__________.【典例2】已知函數(shù)SKIPIF1<0SKIPIF1<0的最大值為SKIPIF1<0SKIPIF1<0的最小值為SKIPIF1<0，則SKIPIF1<0______．【變式1】（多選）設(shè)函數(shù)SKIPIF1<0，SKIPIF1<0存在最小值時(shí)，實(shí)數(shù)SKIPIF1<0的值可能是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．0 D．1題型10分段函數(shù)（解分段不等式）【典例1】設(shè)SKIPIF1<0，則不等式SKIPIF1<0的解集是（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】函數(shù)SKIPIF1<0，若關(guān)于SKIPIF1<0的不等式SKIPIF1<0的解集___________.【變式1】已知SKIPIF1<0,則使SKIPIF1<0成立的SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的最小值；(2)若SKIPIF1<0對(duì)任意SKIPIF1<0恒成立，求k的取值范圍.題型11函數(shù)圖象（函數(shù)圖象識(shí)別）【典例1】設(shè)SKIPIF1<0均為非零實(shí)數(shù)，則直線SKIPIF1<0和SKIPIF1<0在同一坐標(biāo)系下的圖形可能是（

）．A．B．C．D．【典例2】函數(shù)SKIPIF1<0的圖象大致形狀是（

）A．B．C．D．【變式1】已知函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的圖像是（

）A．B．C．D．題型12函數(shù)圖象（畫(huà)出具體函數(shù)圖象）【典例1】給定函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.(1)在所給坐標(biāo)系（1）中畫(huà)出函數(shù)SKIPIF1<0，SKIPIF1<0的大致圖象；（不需列表，直接畫(huà)出.）(2)SKIPIF1<0，用SKIPIF1<0表示SKIPIF1<0，SKIPIF1<0中的較小者，記為SKIPIF1<0，請(qǐng)分別用解析法和圖象法表示函數(shù)SKIPIF1<0.（SKIPIF1<0的圖象畫(huà)在坐標(biāo)系（2）中）(3)直接寫(xiě)出函數(shù)SKIPIF1<0的值域.【典例2】已知函數(shù)SKIPIF1<0.（1）若SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的值；（2）畫(huà)出函數(shù)的圖象并寫(xiě)出函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的值域；（3）若函數(shù)SKIPIF1<0，求函數(shù)SKIPIF1<0在SKIPIF1<0上最大值.【變式1】已知函數(shù)SKIPIF1<0，其中[x]表示不超過(guò)SKIPIF1<0的最大整數(shù)，例如SKIPIF1<0(1)將SKIPIF1<0的解析式寫(xiě)成分段函數(shù)的形式;(2)請(qǐng)?jiān)谌鐖D所示的平面直角坐標(biāo)系中作出函數(shù)SKIPIF1<0的圖象;(3)根據(jù)圖象寫(xiě)出函數(shù)SKIPIF1<0的值域.題型13函數(shù)圖象（根據(jù)實(shí)際問(wèn)題做出函數(shù)圖象）【典例1】如圖為某無(wú)人機(jī)飛行時(shí)，從某時(shí)刻開(kāi)始15分鐘內(nèi)的速度SKIPIF1<0（單位：米/分鐘）與時(shí)間SKIPIF1<0（單位：分鐘）的關(guān)系．若定義“速度差函數(shù)”SKIPIF1<0為無(wú)人機(jī)在時(shí)間段SKIPIF1<0內(nèi)的最大速度與最小速度的差，則SKIPIF1<0的圖像為（

）A．B．C．D．【典例2】如圖，點(diǎn)SKIPIF1<0在邊長(zhǎng)為1的正方形的邊上運(yùn)動(dòng)，SKIPIF1<0是SKIPIF1<0的中點(diǎn)，則當(dāng)SKIPIF1<0沿SKIPIF1<0運(yùn)動(dòng)時(shí)，點(diǎn)SKIPIF1<0經(jīng)過(guò)的路程SKIPIF1<0與SKIPIF1<0的面積SKIPIF1<0的函數(shù)SKIPIF1<0的圖象大致是下圖中的A．B．C．D．【變式1】如圖，公園里有一處扇形花壇，小明同學(xué)從SKIPIF1<0點(diǎn)出發(fā)，沿花壇外側(cè)的小路順時(shí)針?lè)较騽蛩僮吡艘蝗?路線為SKIPIF1<0)，則小明到SKIPIF1<0點(diǎn)的直線距離SKIPIF1<0與他從SKIPIF1<0點(diǎn)出發(fā)后運(yùn)動(dòng)的時(shí)間SKIPIF1<0之間的函數(shù)圖象大致是（

）A．B．C．D．題型14函數(shù)圖象（函數(shù)圖象的變換）【典例1】（多選）下列函數(shù)圖像經(jīng)過(guò)變換后，過(guò)原點(diǎn)的是（

）A．SKIPIF1<0向右平移SKIPIF1<0個(gè)單位 B．SKIPIF1<0向左平移SKIPIF1<0個(gè)單位C．SKIPIF1<0向上平移SKIPIF1<0個(gè)單位 D．SKIPIF1<0向下平移SKIPIF1<0個(gè)單位【典例2】已知函數(shù)SKIPIF1<0定義在SKIPIF1<0上的圖象如圖所示，請(qǐng)分別畫(huà)出下列函數(shù)的圖象：

(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0；(4)SKIPIF1<0；(5)SKIPIF1<0；(6)SKIPIF1<0.【變式1】將函數(shù)SKIPIF1<0的圖象向左平移1個(gè)單位，再向下平移3個(gè)單位長(zhǎng)度，所得的函數(shù)圖象對(duì)應(yīng)的解析式為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型15函數(shù)圖象（根據(jù)圖象選擇解析式）【典例1】我國(guó)著名數(shù)學(xué)家華羅庚曾說(shuō)：“數(shù)缺形時(shí)少直觀，形缺數(shù)時(shí)難入微，數(shù)形結(jié)合百般好，隔裂分家萬(wàn)事休．”在數(shù)學(xué)的學(xué)習(xí)和研究中，常用函數(shù)的圖象來(lái)研究函數(shù)的性質(zhì)，也常用函數(shù)的解析式來(lái)研究函數(shù)圖象的特征．我們從這個(gè)商標(biāo)中抽象出一個(gè)圖象如圖，其對(duì)應(yīng)的函數(shù)可能是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】若函數(shù)SKIPIF1<0的大致圖象如圖所示，則SKIPIF1<0的解析式可能是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式1】已知某函數(shù)SKIPIF1<0的部分圖象如圖所示，則下列函數(shù)解析式符合該圖象特征的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0題型16新定義（新文化題）【典例1】十九世紀(jì)德國(guó)數(shù)學(xué)家狄利克雷提出了“狄利克雷函數(shù)”SKIPIF1<0它在現(xiàn)代數(shù)學(xué)的發(fā)展過(guò)程中有著重要意義，若函數(shù)SKIPIF1<0，則下列實(shí)數(shù)不屬于函數(shù)SKIPIF1<0值域的是（

）A．3 B．2 C．1 D．0【典例2】黎曼函數(shù)是一個(gè)特殊的函數(shù)，由德國(guó)著名的數(shù)學(xué)家波恩哈德·黎曼發(fā)現(xiàn)提出，在高等數(shù)學(xué)中有著廣泛的應(yīng)用.其定義黎曼函數(shù)SKIPIF1<0為：當(dāng)SKIPIF1<0（SKIPIF1<0為正整數(shù)，SKIPIF1<0是既約真分?jǐn)?shù)）時(shí)SKIPIF1<0，當(dāng)SKIPIF1<0或SKIPIF1<0或SKIPIF1<0為SKIPIF1<0上的無(wú)理數(shù)時(shí)SKIPIF1<0.已知SKIPIF1<0、SKIPIF1<0、SKIPIF1<0都是區(qū)間SKIPIF1<0內(nèi)的實(shí)數(shù)，則下列不等式一定正確的是A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式1】高斯是德國(guó)著名的數(shù)學(xué)家，近代數(shù)學(xué)奠基者之一，享有“數(shù)學(xué)王子”的稱號(hào)，他和阿基米德，牛頓并列為世界三大數(shù)學(xué)家，用其名字命名的“高斯函數(shù)”為：對(duì)于實(shí)數(shù)SKIPIF1<0，符號(hào)SKIPIF1<0表示不超過(guò)SKIPIF1<0的最大整數(shù)，則SKIPIF1<0稱為高斯函數(shù)，例如SKIPIF1<0，SKIPIF1<0，定義函數(shù)SKIPIF1<0，則下列命題中正確的序號(hào)是________．①函數(shù)SKIPIF1<0的最大值為SKIPIF1<0；

②函數(shù)SKIPIF1<0的最小值為SKIPIF1<0；③函數(shù)SKIPIF1<0的圖象與直線SKIPIF1<0有無(wú)數(shù)個(gè)交點(diǎn)

④SKIPIF1<0題型17重點(diǎn)方法（換元法）【典例1】若SKIPIF1<0，則SKIPIF1<0＝________.【典例2】已知SKIPIF1<0，則SKIPIF1<0______．【變式2】已知SKIPIF1<0，則SKIPIF1<0___________.題型18重點(diǎn)方法（消去法）【典例1】若函數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0（

）A．0 B．2 C．3 D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0__________.【變式1】已知SKIPIF1<0，則函數(shù)SKIPIF1<0的解析式為_(kāi)__________.題型19數(shù)學(xué)思想方法（數(shù)形結(jié)合法）【典例1】（多選