Question

對(duì)于定義在R上的函數(shù)y=f(x),有下述四個(gè)命題:

①若y=f(x)是奇函數(shù),則y=f(x-1)的圖象關(guān)于點(diǎn)A(1,0)對(duì)稱;

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

③若函數(shù)y=f(x-1)的圖象關(guān)于直線x=1對(duì)稱,則y=f(x)為偶函數(shù);

④函數(shù)y=f(1+x)與函數(shù)y=f(1-x)的圖象關(guān)于x=1對(duì)稱.其中正確命題的個(gè)數(shù)為

A.1                   B.2                  C.3                     D.4

Answer 1

答案:B

解析：①中f(x)的圖象關(guān)于原點(diǎn)對(duì)稱,則f(x-1)的圖象關(guān)于(1,0)點(diǎn)對(duì)稱;②中f(x+1)=f(x-1)時(shí),滿足f(3)=f(1),又函數(shù)f(x)不恒為常函數(shù),∴圖象不一定關(guān)于x=1對(duì)稱;③∵f(x-1)關(guān)于x=1對(duì)稱,則f(x)的圖象關(guān)于y軸對(duì)稱,

∴f(x)為偶函數(shù);④當(dāng)x=1時(shí),兩函數(shù)的函數(shù)值分別為y₁=f(2),y₂=f(0),且y₁≠y₂.

綜上知①③正確.