Question

函數(shù)y=f(x)(x∈R)有下列命題:
①在同一坐標(biāo)系中,y=f(x+1)與y=f(-x+1)的圖象關(guān)于直線x=1對(duì)稱;
②若f(2-x)=f(x),則函數(shù)y=f(x)的圖象關(guān)于直線x=1對(duì)稱;
③若f(x-1)=f(x+1),則函數(shù)y=f(x)是周期函數(shù),且2是一個(gè)周期;
④若f(2-x)=-f(x),則函數(shù)y=f(x)的圖象關(guān)于(1,0)對(duì)稱,其中正確命題的序號(hào)是　　　　.

Answer 1

②③④

對(duì)于①,y=f(x+1)的圖象由y=f(x)的圖象向左平移1個(gè)單位得到,y=f(-x+1)的圖象,由y=f(-x)的圖象向右平移1個(gè)單位得到,而y=f(x)與y=f(-x)關(guān)于y軸對(duì)稱,從而y=f(x+1)與y=f(-x+1)的圖象關(guān)于直線x=0對(duì)稱,故①錯(cuò);
對(duì)于②,由f(2-x)=f(x)將x換為x+1可得f(1-x)=f(1+x),從而②正確;
對(duì)于③,由f(x-1)=f(x+1)將x換為x+1可得,f(x+2)=f(x),從而③正確.
對(duì)于④,由f(2-x)=-f(x)同上可得f(1-x)=-f(1+x),從而④正確.
【誤區(qū)警示】解答本題時(shí),易誤以為①正確,出錯(cuò)的原因是混淆了兩個(gè)函數(shù)y=f(x+1)與y=f(-x+1)的圖象關(guān)系與一個(gè)函數(shù)y=f(x)滿足f(x+1)=f(-x+1)時(shí)圖象的對(duì)稱關(guān)系.