一般地，對(duì)于函數(shù)y=f(x(，如果存在一個(gè)非零常數(shù)T，使得當(dāng)x取定義域內(nèi)的每一個(gè)值時(shí)，f(x+T)=正周期.①若函數(shù)f(x)滿足f(x+a)=f(x)，則函數(shù)f(x)的周期T=a.②若函數(shù)f(x)滿足f(x+a)=f(x-a)，則函數(shù)f(x)的周期T=2a.③若函數(shù)f(x)滿足f(x+a)=-f(x)，則函數(shù)f(x)的周期T=2a.④若函數(shù)f(x)滿足f，則函數(shù)f(x)的周期T=2a.⑤若函數(shù)f(x)滿足f，則函數(shù)f(x)的周期T=2a.f(x)=5-2x，則f)．A.B.CDAB.0C.2D.-13.(25-26高三上·江蘇南通·開學(xué)考試)定義在R上的函數(shù)f(x)是周期為2的偶函數(shù)，當(dāng)0≤x≤1時(shí)，A.2B.1C.-2D.-1A.-1B.0C.1D.2f(x)=5-2x，則f(5)=()A.-5B.-1C.1D.46.(25-26高三上·海南省直轄縣級(jí)單位·階段練習(xí))已知定義在R上的函數(shù)y=f(x(滿足f(x(=f(x+4(，7.(24-25高一下·貴州黔南·階段練習(xí))若函數(shù)y=f(x)是R上的周期為8.(25-26高三上·福建龍巖·階段練習(xí))已知定義在R上的f(x)函數(shù)x ①若函數(shù)f(x)滿足f(a+x)=f(a-x)，則函數(shù)f(x)關(guān)于x=a對(duì)稱.②若函數(shù)f(x)滿足f(a+x)+f(a-x)=0，則函數(shù)f(x)關(guān)于(a,0)對(duì)稱.③若函數(shù)f(x)滿足f(a+x)=f(b-x)，則函數(shù)f(x)關(guān)于x=對(duì)稱.④若函數(shù)f(x)滿足f(a+x)+f(b-x)=0，則函數(shù)f(x)關(guān)于,0(對(duì)稱.⑤若函數(shù)f(x)滿足f(a+x)+f(b-x)=m，則函數(shù)f(x)關(guān)于,對(duì)稱.A.3B.4C.-3D.-4()A.-3B.3C.-6D.6()A.-5B.-1C.0D.5A.關(guān)于點(diǎn)(-2,2(對(duì)稱B.關(guān)于點(diǎn)(2,-2(對(duì)稱C.關(guān)于直線x=2對(duì)稱D.關(guān)于直線x=-2對(duì)稱 “?x∈D，f(2m-x)+f(x)= (1)求f(0(與f(2)，f(-1(與f(3)的值；(2)由(1)中求得的結(jié)果，猜想f(x(與f(2-x(的關(guān)系并證明你的猜想；(3)求f(-2020(+f(-2019(+...+f(0(+f(1)+f(2)+...+f(2021(+f(2022(的值．核心定義奇函數(shù)：f(-x(=-f(x(偶函數(shù)：f(-x(=f(x(存在非零常數(shù)T，使f(x+T(=f(x(，最小的正T稱為“最小正周期”f(a+x(=f(a-x(f(a+x(+f(a-x(=2b23.(25-26高二上·湖北武漢·階段練習(xí))已知函數(shù)f(x),M(x)的定義域?yàn)镽，滿足M(4-x)+f(x)=9,MA.92B.-205C.100D.-1924.(25-26高三上·江蘇揚(yáng)州·開學(xué)考試)已知f(x(是定義在R上的奇函數(shù)，函數(shù)g(x(=(x-1(f(x(的圖象關(guān)于點(diǎn)(1,0(對(duì)稱，且滿足g(-1(=4，則f(A.2B.-4C.2026D.-4052，若f(0(+f(3(=6，則y=f(x(-m(0<m<6(在x∈(0,8(內(nèi)所有的零26.(25-26高三上·廣東·開學(xué)考試)已知函數(shù)f(x(的定義域?yàn)镽，函數(shù)y=f(x+1(是偶函數(shù)，函數(shù)y= A.-1B.0C.1D.2A.f(2(=0B.f(x(的一個(gè)周期為429.(25-26高三上·四川綿陽·階段練習(xí)·多選)定義在R上的函數(shù)f(x(滿足f(1+x(=f(1-x(，且f(x+2(A.f(x+4(=f(x(B.f(x(在區(qū)間[9,11[上單調(diào)遞減30.(25-26高三上·河北保定·階段練習(xí)·多選)已知f(x(，g(x(均為定義域?yàn)镽的奇函數(shù)，且f(x(+31.(25-26高三上·寧夏銀川·開學(xué)考試·多選)已知定義域?yàn)镽的函數(shù)f(x(滿足f(4-3x(=f(3x-2(,f(4x-1(為奇函數(shù)，f(0(=-1，則()A.f(x(是周期為8的函數(shù)B.f(x-3(為偶函數(shù)C.f(1(+f(5(=1D.A.函數(shù)f(x)的圖象關(guān)于x=-1對(duì)稱B.g(x)是周期為4的函數(shù)CD.y=g(x+1(是奇函數(shù)33.(25-26高三上·陜西漢中·階段練習(xí)·多選)已知函數(shù)f(x(，g(x(的定義域均為R，f(x+3(g(x(關(guān)于直線x=-3對(duì)稱，且f(x(+g(-x-3(=2，若g(-3(=1，則()A.f(0(=1B.f(x(的圖象關(guān)于點(diǎn)中心對(duì)稱C.f(x(是偶函數(shù)D.f(0(+f(1(+f(2(+...+f(2025(=-134.(25-26高三上·河北衡水·階段練習(xí)·多選)已知函數(shù)f(x(是定義在R上的奇函數(shù)，f(x+1(是偶函數(shù)，A.函數(shù)f(x(關(guān)于直線x=1對(duì)稱B.4是函數(shù)f(x(的周期C.f(2025(+f(2026(=0D.方程f(x(=|lnx|恰有4不同的根36.(24-25高二下·吉林白山·期末)已知f(x+1(是定義在R上的奇函數(shù)，且f(x+4(=f(2-x(，當(dāng)x∈37.(25-26高三上·山東濰坊·階段練習(xí))已知函數(shù)f(x(的定義域?yàn)镽，且滿足f(1+x(=-f(1-x(,f(x+2(f(x(=ax2+bx，若f(0(+f(3(=38.(25-26高三上·山東臨沂·階段練習(xí))已知函數(shù)f(x(的定義域?yàn)镽，且f(x+2(-2為奇函數(shù)，f(3x+1( 1.(2025·湖南·一模)已知f(x(是定義在R上的奇函數(shù)，f(x+1(為偶函數(shù)，且當(dāng)x∈[0,1[時(shí)，f(x(=sinx，則函數(shù)g(x(=f(x在[-5,4)∪(4,13[上所有零點(diǎn)的和為()A.f(-3)=0B.f(0)=0C.f(2)=0D.f(4)=03.(25-26高三上·江西·階段練習(xí))已知函數(shù)f(x)滿足f(x+1)-f(x-1)=3，且y=f(x+2)+1為奇函4.(25-26高三上·新疆·階段練習(xí))定義在R上的奇函數(shù)f(x(滿足f(x(=f(2-x(，當(dāng)x∈(0,1[時(shí)，f(x(=2x，則f(2026(-f(2025(的值為()A.logB.1C.-1D.-25.(25-26高三上·山西·階段練習(xí))已知函數(shù)f(x(的定義域?yàn)镽，其圖象關(guān)于直線x=-3對(duì)稱且f(x+3(A.函數(shù)f(x(為偶函數(shù)B.函數(shù)f(x(在[-6,-3[上單調(diào)遞增C.函數(shù)f(x(的圖象關(guān)于直線x=3對(duì)稱D.f(2026(=-76.(25-26高三上·福建莆田·階段練習(xí))定義在R時(shí)，恒有f(x1(≤f(x2(，則f)ABCD7.(25-26高三上·安徽阜陽·階段練習(xí)·多選)已知定義在R上的可導(dǎo)函數(shù)f(x(的導(dǎo)函數(shù)為g(x(，若f(2x+2(為奇函數(shù)，g(x+1(+g(1-x(=2，則()A.函數(shù)f(x(的圖象關(guān)于點(diǎn)(2,0(對(duì)稱B.函數(shù)g(x(的圖象關(guān)于直線x=5對(duì)稱C.4是函數(shù)g(x(的一個(gè)周期D.g(2k-1(=2008.(2025·湖北黃岡·一?！ざ噙x)定義在R上的函數(shù)f(x(和g(x(，f(x+2(為奇函數(shù)，g(x(為偶函數(shù)，且f(x+1(+g(3-x(=4，則()A.g(2(=2B.f(6(=0C.f(x(的圖象關(guān)于x=4對(duì)稱D.8為g(x(的一個(gè)周期9.(25-26高三上·四川綿陽·階段練習(xí)·多選)已知函數(shù)f(x)定義域?yàn)镽，其導(dǎo)函數(shù)為g(x)，且f(2-x)+A.f(x)一個(gè)對(duì)稱中心為(1,1)B.g(x)的一個(gè)周期為210.(25-26高三上·河南商丘·開學(xué)考試·多選)若f(x+1(為奇函數(shù)，且f(3-x(=f(1+x(，則下列說法正確A.f(1(=0B.f(x(的一個(gè)周期為2C.f(x-4(=f(x(D.f(4k+1(=011.(25-26高三上·江蘇淮安·階段練習(xí)·多選)設(shè)定義在R上的函數(shù)f(x(與g(x(的導(dǎo)函數(shù)分別為f/(x(和，且f(x+2(=g(2-x(+2,f/(x(=g/(x+2(，且g(x(的圖象關(guān)于點(diǎn)(1,0(對(duì)稱，則()A.f(3(=3B.g/(2-x(+g/(x+4(=0C.g(4-x(=g(x+