抽象函數(shù)性質(zhì)

1、抽象函數(shù)性質(zhì)概述抽象函數(shù)型綜合問題一般通過函數(shù)代數(shù)表示，綜合考察學(xué)生對(duì)數(shù)學(xué)符號(hào)語言的理解和接受能力，考察函數(shù)代數(shù)推論和論證能力，考察學(xué)生對(duì)一般和特殊關(guān)系的認(rèn)識(shí)函數(shù)的周期性、對(duì)稱性一般與抽象函數(shù)結(jié)合，綜合考察函數(shù)的其他性質(zhì)函數(shù)的周期性必須壓迫周期函數(shù)的定義。 應(yīng)該注意的是，函數(shù)的周期性只與一個(gè)函數(shù)相關(guān)函數(shù)的對(duì)稱性比較復(fù)雜，必須區(qū)分一個(gè)函數(shù)的對(duì)稱性還是兩個(gè)函數(shù)的對(duì)稱性，區(qū)分軸對(duì)稱還是中心對(duì)稱一、基本定義1、定義1:(周期函數(shù))函數(shù)，如取定義域的各值時(shí)那樣，在有零以外的常數(shù)的情況下，函數(shù)被稱為周期函數(shù)，零以外的常數(shù)被稱為該函數(shù)的周期.2、定義2:(同一函數(shù)圖像的對(duì)稱性)如果函數(shù)圖像的任一點(diǎn)關(guān)于點(diǎn)(或

2、直線)對(duì)稱，則函數(shù)圖像關(guān)于點(diǎn)(或直線)對(duì)稱.3、定義3:(兩個(gè)函數(shù)圖像的對(duì)稱性)關(guān)于函數(shù)圖像的任一點(diǎn)點(diǎn)(或直線)的對(duì)稱點(diǎn)在函數(shù)的圖像上的情況下，相反，函數(shù)圖像的任意點(diǎn)關(guān)于點(diǎn)(或直線)的對(duì)稱點(diǎn)也在函數(shù)的圖像上，被稱為函數(shù)的圖像關(guān)于點(diǎn)(或直線)對(duì)稱.二、關(guān)于周期性、對(duì)稱性的幾個(gè)基本結(jié)論和證明1 .函數(shù)的定義域中，如果常數(shù)成立，則函數(shù)是周期性的周期函數(shù)2 .如果函數(shù)的定義域大，且常數(shù)成立，則函數(shù)的圖像關(guān)于直線對(duì)稱3 .如果函數(shù)的定義域中，常數(shù)成立，則函數(shù)的圖像是點(diǎn)對(duì)稱的4 .如果函數(shù)的定義域中，常數(shù)成立，則函數(shù)是周期性的周期函數(shù)5 .如果函數(shù)的定義域是定義域，則該圖像和函數(shù)關(guān)于直線對(duì)稱6 .如果函數(shù)

3、的定義域是定義域，則該圖像和函數(shù)是點(diǎn)對(duì)稱的略證： 1、函數(shù)是周期性的周期函數(shù)2 .函數(shù)圖像上關(guān)于任意點(diǎn)(滿意)直線的對(duì)稱點(diǎn)是因?yàn)辄c(diǎn)還在函數(shù)的圖像上，所以函數(shù)的圖像關(guān)于直線對(duì)稱3 .關(guān)于函數(shù)圖像上任意點(diǎn)(滿足)點(diǎn)的對(duì)稱點(diǎn)是因?yàn)辄c(diǎn)還在函數(shù)的圖像上，所以函數(shù)的圖像關(guān)于點(diǎn)是對(duì)稱的4、函數(shù)是周期性的周期函數(shù)5 .函數(shù)圖像上關(guān)于任意點(diǎn)(滿意)直線的對(duì)稱點(diǎn)是與在函數(shù)的圖像上放置點(diǎn)相反，函數(shù)的圖像的任何點(diǎn)關(guān)于直線也在函數(shù)圖像上，所以該圖像與函數(shù)關(guān)于直線對(duì)稱6 .關(guān)于函數(shù)圖像上任意點(diǎn)(滿足)點(diǎn)的對(duì)稱點(diǎn)是與將點(diǎn)放置在函數(shù)的圖像上相反，函數(shù)的圖像的任意點(diǎn)也處于關(guān)于點(diǎn)的對(duì)稱點(diǎn)，所以該圖像與函數(shù)關(guān)于點(diǎn)對(duì)稱三、關(guān)于周期性

4、、對(duì)稱性的幾個(gè)容易混淆的共同結(jié)論1 .如果滿足函數(shù)，則函數(shù)的圖像關(guān)于軸對(duì)稱，函數(shù)和函數(shù)的圖像也軸對(duì)稱2 .如果滿足函數(shù)，則函數(shù)的圖像關(guān)于原點(diǎn)對(duì)稱，而函數(shù)的圖像也關(guān)于原點(diǎn)對(duì)稱3 .如果滿足函數(shù)，則關(guān)于軸對(duì)稱地使函數(shù)的圖像與函數(shù)的圖像關(guān)于直線對(duì)稱4 .如果滿足函數(shù)，則函數(shù)的圖像關(guān)于原點(diǎn)對(duì)稱，而函數(shù)和函數(shù)的圖像關(guān)于點(diǎn)對(duì)稱。5 .如果函數(shù)滿足，函數(shù)圖像關(guān)于直線對(duì)稱函數(shù)和函數(shù)圖像是軸對(duì)稱的6 .如果滿足函數(shù)，則該函數(shù)的圖像是關(guān)于點(diǎn)對(duì)稱的，而該函數(shù)的圖像是關(guān)于原點(diǎn)對(duì)稱的7 .如果滿足函數(shù)，則函數(shù)的圖像關(guān)于直線對(duì)稱，而函數(shù)的圖像也關(guān)于直線對(duì)稱8 .如果滿足函數(shù)，則函數(shù)的圖像是點(diǎn)對(duì)稱的函數(shù)和函數(shù)的圖像也是點(diǎn)對(duì)

5、稱的9 .若函數(shù)滿足，則函數(shù)是周期性周期函數(shù)，若函數(shù)滿足，則函數(shù)是周期性周期函數(shù)四、函數(shù)的周期性和對(duì)稱性的關(guān)系1、以上定義的函數(shù)是針對(duì)直線和對(duì)稱的，即，對(duì)于任何實(shí)數(shù)，如果函數(shù)同時(shí)滿足，則該函數(shù)是周期性的周期函數(shù)2、以上定義的函數(shù)是點(diǎn)和點(diǎn)的對(duì)稱性，即，如果函數(shù)同時(shí)對(duì)任意實(shí)數(shù)滿足，則該函數(shù)是周期性的周期函數(shù)3、以上定義的函數(shù)是線性和點(diǎn)對(duì)稱的，即，如果該函數(shù)同時(shí)對(duì)任何實(shí)數(shù)滿足，則該函數(shù)是周期性的周期函數(shù)證書：1、=函數(shù)是周期性的周期函數(shù)兩三件事都可以證明五、函數(shù)的周期性、對(duì)稱性與奇偶的關(guān)系1、以上定義的函數(shù)是針對(duì)直線和對(duì)稱的，即對(duì)于任意實(shí)數(shù)，如果函數(shù)同時(shí)滿足，則該函數(shù)是周期性的周期函數(shù)，是偶然函數(shù)2

6、、以上定義的函數(shù)是線性和點(diǎn)對(duì)稱的，即，是奇數(shù)函數(shù)，如果該函數(shù)同時(shí)對(duì)任何實(shí)數(shù)滿足，則該函數(shù)是周期性的3、以上定義的函數(shù)是關(guān)于點(diǎn)和線對(duì)稱的，即，如果函數(shù)同時(shí)對(duì)任何實(shí)數(shù)滿足，則該函數(shù)是周期性的周期函數(shù)，并且是偶函數(shù)4、以上定義的函數(shù)如果同時(shí)對(duì)點(diǎn)和點(diǎn)的對(duì)稱性，即，對(duì)任意的實(shí)數(shù)滿足函數(shù)，則該函數(shù)為周期性的周期函數(shù)，并且是奇函數(shù)5、偶發(fā)函數(shù)關(guān)于直線對(duì)稱，即，如果對(duì)任意實(shí)數(shù)滿足函數(shù)，則是周期性的周期函數(shù).6 .如果偶發(fā)函數(shù)對(duì)點(diǎn)對(duì)稱，即對(duì)任意實(shí)數(shù)滿足函數(shù)，則是周期性的周期函數(shù)7 .奇函數(shù)關(guān)于直線對(duì)稱，即，如果對(duì)任何實(shí)數(shù)滿足函數(shù)，那么該奇函數(shù)為周期性函數(shù)8 .若奇函數(shù)是點(diǎn)對(duì)稱的，即，對(duì)任意實(shí)數(shù)滿足函數(shù)，則是周期性的周期函數(shù)證書：1 .在上述四個(gè)點(diǎn)中的第一點(diǎn)處獲得的函數(shù)是周期性的周期函數(shù)又來了函數(shù)是偶函數(shù)2、3、4同樣地可以證明. 5、6、7、8可以利用上述4個(gè)結(jié)論來證明.以上各結(jié)論可以作為特例理解正弦、馀弦函數(shù).六、其他結(jié)論1 .如果函數(shù)是偶函數(shù)，則該函數(shù)的圖像關(guān)于直線對(duì)稱2 .如果函數(shù)是奇函數(shù)，則該函數(shù)的圖像是點(diǎn)對(duì)稱的注：上述兩個(gè)結(jié)論可以通過圖像的移位來理解3、定義上的函數(shù)滿足，方程式有實(shí)根時(shí)，該實(shí)根之和為4、如果滿足定義上的函數(shù)，則函數(shù)的圖像是點(diǎn)對(duì)稱的略證任取、令、則通過中點(diǎn)式知道點(diǎn)和點(diǎn)關(guān)于點(diǎn)對(duì)稱.的任意性，知道函數(shù)的圖像關(guān)于點(diǎn)對(duì)稱.5、函數(shù)是周期函數(shù)的常見結(jié)論是，函數(shù)