高一數(shù)學(xué)必修一第二章基本初等函數(shù)知識(shí)點(diǎn)總結(jié)

1、2.1指數(shù)函數(shù)2.1.1指數(shù)與指數(shù)冪的運(yùn)算(1根式的概念 如果,1nxa a R x R n =>,且n N +,那么x 叫做a 的n 次方根.當(dāng)n 是奇數(shù)時(shí),a 的n 表示;當(dāng)n 是偶數(shù)時(shí),正數(shù)a 的正的nn次方根用符號(hào)0的n 次方根是0;負(fù)數(shù) a 沒有n 次方根.n 叫做根指數(shù),a 叫做被開方數(shù).當(dāng)n 為奇數(shù)時(shí),a 為任意實(shí)數(shù);當(dāng)n 為偶數(shù)時(shí),0a . 根式的性質(zhì):n a =;當(dāng)na =;當(dāng)n 為偶數(shù)時(shí),(0| (0a a a a a = -<. (2分?jǐn)?shù)指數(shù)冪的概念正數(shù)的正分?jǐn)?shù)指數(shù)冪的意義是:0,m naa m n N +=>且1n >.0的正分?jǐn)?shù)指數(shù)冪等于0.正

2、數(shù)的負(fù)分?jǐn)?shù) 指數(shù)冪的意義是: 1(0,m m nn aa m n N a -+=>且1n >.0的負(fù)分?jǐn)?shù)指數(shù)冪沒有意義. 注意口訣:底 數(shù)取倒數(shù),指數(shù)取相反數(shù). (3分?jǐn)?shù)指數(shù)冪的運(yùn)算性質(zhì)(0,rs r s aa a a r s R +=> (0,r s rs a a a r s R => (0,0,r r r ab a b a b r R =>>2.1.2指數(shù)函數(shù)及其性質(zhì)(4指數(shù)函數(shù) 2.2對(duì)數(shù)函數(shù)【2.2.1】對(duì)數(shù)與對(duì)數(shù)運(yùn)算(1對(duì)數(shù)的定義若(0,1xaN a a =>且,則x 叫做以a 為底N 的對(duì)數(shù),記作log a xN =,其中a 叫做底數(shù),N叫

3、做真數(shù).負(fù)數(shù)和零沒有對(duì)數(shù).對(duì)數(shù)式與指數(shù)式的互化:log (0,1,0x a x N a N a a N =>>.(2幾個(gè)重要的對(duì)數(shù)恒等式: log 10a =,log 1aa =,logb a a b =. (3常用對(duì)數(shù)與自然對(duì)數(shù):常用對(duì)數(shù):lg N ,即10log N ;自然對(duì)數(shù):ln N,即log eN (其中 2.71828e =. (4對(duì)數(shù)的運(yùn)算性質(zhì) 如果0,1,0,0aa M N >>>,那么加法:log log log (aa a M N MN += 減法:log log log a a aMM N N-=數(shù)乘:log log (na a n M M

4、n R = log a NaN =log log (0,b na a n M Mb n R b= 換底公式:log log (0,1log b a b N N b b a =>且【2.2.2】對(duì)數(shù)函數(shù)及其性質(zhì)(5對(duì)數(shù)函數(shù) (6反函數(shù)的概念設(shè)函數(shù)(y f x =的定義域?yàn)锳 ,值域?yàn)镃 ,從式子(y f x =中解出x ,得式子(x y =.如果對(duì)于y 在C 中的任何一個(gè)值,通過式子(x y =,x 在A 中都有唯一確定的值和它對(duì)應(yīng),那么式子(x y =表示x 是y 的函數(shù),函數(shù)(xy =叫做函數(shù)(y f x =的反函數(shù),記作1(x f y -=,習(xí)慣上改寫成1(y f x -=.(7反函

5、數(shù)的求法確定反函數(shù)的定義域,即原函數(shù)的值域;從原函數(shù)式(y f x =中反解出1(x f y -=;將1(xf y -=改寫成1(y f x -=,并注明反函數(shù)的定義域.(8反函數(shù)的性質(zhì)原函數(shù)(y f x =與反函數(shù)1(y f x -=的圖象關(guān)于直線y x =對(duì)稱.函數(shù)(y f x =的定義域、值域分別是其反函數(shù)1(y f x -=的值域、定義域.若(,P a b 在原函數(shù)(y f x =的圖象上,則'(,P b a 在反函數(shù)1(y f x -=的圖象上.一般地,函數(shù)(y f x =要有反函數(shù)則它必須為單調(diào)函數(shù).2.3冪函數(shù)(1冪函數(shù)的定義 一般地,函數(shù)y x =叫做冪函數(shù),其中x 為

6、自變量,是常數(shù).(2冪函數(shù)的圖象 (3冪函數(shù)的性質(zhì)圖象分布:冪函數(shù)圖象分布在第一、二、三象限,第四象限無(wú)圖象.冪函數(shù)是偶函數(shù)時(shí),圖象分布在第一、二象限(圖象關(guān)于y 軸對(duì)稱;是奇函數(shù)時(shí),圖象分布在第一、三象限(圖象關(guān)于原點(diǎn)對(duì)稱;是非奇非偶函數(shù)時(shí),圖象只分布在第一象限.過定點(diǎn):所有的冪函數(shù)在(0,+都有定義,并且圖象都通過點(diǎn)(1,1. 單調(diào)性:如果0>,則冪函數(shù)的圖象過原點(diǎn),并且在0,+上為增函數(shù).如果0<,則冪函數(shù)的圖象在(0,+上為減函數(shù),在第一象限內(nèi),圖象無(wú)限接近x 軸與y 軸.奇偶性:當(dāng)為奇數(shù)時(shí),冪函數(shù)為奇函數(shù),當(dāng)為偶數(shù)時(shí),冪函數(shù)為偶函數(shù).當(dāng)qp=(其中,p q 互質(zhì),p 和q Z ,若p 為奇數(shù)q 為奇數(shù)時(shí),則qpy x =是奇函數(shù),若p 為奇數(shù)q 為偶數(shù)時(shí),則qpy x =是偶函數(shù),若p 為偶數(shù)q 為奇數(shù)時(shí),則q py x=是非奇非偶函數(shù).圖象特征:冪函數(shù),(0,y x x =+,當(dāng)1>時(shí),若01x <