2025版新高考版高考總復(fù)習(xí)數(shù)學(xué)函數(shù)的周期性和對稱性

2025版新高考版高考總復(fù)習(xí)數(shù)學(xué)3.3函數(shù)的周期性和對稱性五年高考考點1函數(shù)的周期性1.(2021全國甲文,12,5分,中)設(shè)f(x)是定義域為R的奇函數(shù),且f(1+x)=f(-x).若f?13=13,則f5A.-5答案C(2016山東理,9,5分,中)已知函數(shù)f(x)的定義域為R.當x<0時,f(x)=x3-1;當-1≤x≤1時,f(-x)=-f(x);當x>12時,fx+12=fx?12,A.-2B.-1C.0D.2答案D3.(2022新高考Ⅱ,8,5分,難)已知函數(shù)f(x)的定義域為R,且f(x+y)+f(x-y)=f(x)f(y),f(1)=1,則k=122f(k)=(A.-3B.-2C.0D.1答案A4.(2018江蘇,9,5分,中)函數(shù)f(x)滿足f(x+4)=f(x)(x∈R),且在區(qū)間(-2,2]上,f(x)=cosπx2,0<x≤2,x+12答案25.(2016江蘇,11,5分,中)設(shè)f(x)是定義在R上且周期為2的函數(shù),在區(qū)間[-1,1)上,f(x)=x+a,?1≤x<0,25?x,0≤x<1,其中a∈R.答案-2考點2函數(shù)的對稱性(2022全國乙理,12,5分,難)已知函數(shù)f(x),g(x)的定義域均為R,且f(x)+g(2-x)=5,g(x)-f(x-4)=7.若y=g(x)的圖象關(guān)于直線x=2對稱,g(2)=4,則k=122f(k)=(A.-21B.-22C.-23D.-24答案D三年模擬綜合拔高練1.(2024屆河北保定月考,7)已知定義在R上的函數(shù)f(x)滿足f(x+2)=3f(x),且f(2)=-1,則f(100)=A.-1B.1C.-3D.3答案C2.(2024屆重慶七校開學(xué)考,3)定義在R上的奇函數(shù)f(x)滿足f(x+4)=f(x),當x∈(0,2)時,f(x)=3x-1,則f(2022)+f(2023)=()A.-2023B.-1C.1D.32022答案B3.(2023福建廈門一模,6)定義在R上的函數(shù)f(x)滿足f(x+1)=f(x)-2,則下列是周期函數(shù)的是()A.y=f(x)-xB.y=f(x)+xC.y=f(x)-2xD.y=f(x)+2x答案D4.(2024屆福建連城一中月考,7)已知定義在R上的函數(shù)f(x)滿足①f(x+2)=f(x),②f(x-2)為奇函數(shù),③當x∈[0,1)時,f(x1)?f(x2)x1?x2>0(x1≠x2)恒成立,則f?A.f?152>f(4)>fB.f?152>fC.f112>f(4)>fD.f(4)>f11答案A5.(2024屆遼寧名校聯(lián)盟聯(lián)考,4)若函數(shù)y=f(x),y=g(x)的定義域均為R,且都不恒為零,則()A.若y=f(g(x))為周期函數(shù),則y=g(x)為周期函數(shù)B.若y=f(g(x))為偶函數(shù),則y=g(x)為偶函數(shù)C.若y=f(x),y=g(x)均為單調(diào)遞增函數(shù),則y=f(x)·g(x)為單調(diào)遞增函數(shù)D.若y=f(x),y=g(x)均為奇函數(shù),則y=f(g(x))為奇函數(shù)答案D6.(2024屆山東濟南歷城二中開學(xué)摸底檢測,7)已知函數(shù)f(x)是R上的偶函數(shù),且f(x)的圖象關(guān)于點(1,0)對稱,當x∈[0,1]時,f(x)=2-2x,則f(0)+f(1)+f(2)+…+f(2022)的值為()A.-2B.-1C.0D.1答案C7.(多選)(2024屆河北石家莊月考,12)已知函數(shù)y=xf(x)是R上的偶函數(shù),f(x-1)+f(x+3)=0,當x∈[-2,0]時,f(x)=2x-2-x+x,則()A.f(x)的圖象關(guān)于直線x=2對稱B.4是f(x)的一個周期C.f(x)在(0,2]上單調(diào)遞增D.f(2023)<f12<f(0.50.2答案ACD8.(多選)(2024屆山東部分學(xué)校開學(xué)摸底聯(lián)考,10)已知函數(shù)f(x)對任意x∈R都有f(x)=f(x+4)+f(2),若函數(shù)y=f(x+3)的圖象關(guān)于直線x=-3對稱,且?x1,x2∈[0,2],當x1≠x2時,都有(x2-x1)(f(x2)-f(x1))>0,則下列結(jié)論正確的是()A.f(2)=0B.f(x)是奇函數(shù)C.f(x)是周期為4的周期函數(shù)D.f(3)<f(-4)答案AC9.(多選)(2023湖南師大附中5月模擬,10)已知定義在R上的函數(shù)f(x)滿足f(x+2)+f(x)=0,且y=f(2-x)為偶函數(shù),則下列說法一定正確的是()A.函數(shù)f(x)的周期為2B.函數(shù)f(x)的圖象關(guān)于(1,0)對稱C.函數(shù)f(x)為偶函數(shù)D.函數(shù)f(x)的圖象關(guān)于直線x=3對稱答案BC10.(多選)(2024屆山東臨沂開學(xué)摸底聯(lián)考,12)已知f(x)是定義在R上的奇函數(shù),f(1-2x