Question

對(duì)于定義在R上的函數(shù)f(x)，有下述四個(gè)命題：

①若f(x)是奇函數(shù)，則f(x-1)的圖像關(guān)于點(diǎn)A(1，0)對(duì)稱(chēng)；

②若對(duì)于任意x∈R,有f(x+1)=f(x-1)，則函數(shù)f(x)的圖像關(guān)于直線x=1對(duì)稱(chēng)；

③若函數(shù)f(x-1)的圖像關(guān)于直線x=1對(duì)稱(chēng)，則f(x)為偶函數(shù)；

④函數(shù)y=f(1+x)與函數(shù)y=f(1-x)的圖像關(guān)于直線x=1對(duì)稱(chēng).

其中正確命題的序號(hào)為_(kāi)_________.(把你認(rèn)為正確命題的序號(hào)都填上)

Answer 1

解析：①中，將函數(shù)f(x)的圖像向右平移1個(gè)單位得函數(shù)f(x-1)的圖像，函數(shù)f(x)關(guān)于點(diǎn)(0,0)對(duì)稱(chēng)，則函數(shù)f(x-1)的圖像關(guān)于點(diǎn)(1，0)對(duì)稱(chēng)，則①正確；②中，假設(shè)函數(shù)f(x)的圖像關(guān)于直線x=1對(duì)稱(chēng)，則對(duì)x∈R,有f(x+1)=f(1-x)，但是f(1-x)與f(x-1)不相等，則②錯(cuò)誤；③中，將函數(shù)f(x-1)的圖像向左平移一個(gè)單位得函數(shù)f(x)的圖像，又函數(shù)f(x-1)的圖像關(guān)于直線x=1對(duì)稱(chēng)，則函數(shù)f(x)的圖像關(guān)于y軸對(duì)稱(chēng)，則有函數(shù)f(x)為偶函數(shù)，則③正確；④可以證明函數(shù)y=f(1+x)與函數(shù)y=f(1-x)的圖像關(guān)于直線x=1對(duì)稱(chēng)，則④正確.

答案：①③④