第二章函數(shù)第4節(jié)函數(shù)的對(duì)稱(chēng)性及應(yīng)用學(xué)習(xí)導(dǎo)航站核心知識(shí)庫(kù)：重難考點(diǎn)總結(jié)，梳理必背知識(shí)、歸納重點(diǎn)考點(diǎn)1奇函數(shù)、偶函數(shù)的對(duì)稱(chēng)性★★★☆☆考點(diǎn)2若函數(shù)y=f(x)的圖象關(guān)于直線x=a對(duì)稱(chēng),則f(a-x)=f(a+x)★★★☆☆考點(diǎn)3兩個(gè)函數(shù)圖象的對(duì)稱(chēng)★★★☆☆（星級(jí)越高，重要程度越高）限時(shí)【變式訓(xùn)練】挑戰(zhàn)場(chǎng)：感知真題，檢驗(yàn)成果，考點(diǎn)追溯【知識(shí)梳理】考點(diǎn)1奇函數(shù)、偶函數(shù)的對(duì)稱(chēng)性(1)奇函數(shù)的圖象關(guān)于原點(diǎn)對(duì)稱(chēng),偶函數(shù)的圖象關(guān)于y軸對(duì)稱(chēng).(2)若f(x-2)是偶函數(shù),則函數(shù)f(x)圖象的對(duì)稱(chēng)軸為x=-2;若f(x-2)是奇函數(shù),則函數(shù)f(x)的圖象的對(duì)稱(chēng)中心為(-2,0).考點(diǎn)2若函數(shù)y=f(x)的圖象關(guān)于直線x=a對(duì)稱(chēng),則f(a-x)=f(a+x);若函數(shù)y=f(x)滿足f(a-x)=-f(a+x),則函數(shù)的圖象關(guān)于點(diǎn)(a,0)對(duì)稱(chēng).考點(diǎn)3兩個(gè)函數(shù)圖象的對(duì)稱(chēng)(1)函數(shù)y=f(x)與y=f(-x)關(guān)于y軸對(duì)稱(chēng);(2)函數(shù)y=f(x)與y=-f(x)關(guān)于x軸對(duì)稱(chēng);(3)函數(shù)y=f(x)與y=-f(-x)關(guān)于原點(diǎn)對(duì)稱(chēng).【名師點(diǎn)撥】對(duì)稱(chēng)性的四個(gè)常用結(jié)論(1)y=f(x+a)是偶函數(shù)?f(a+x)=f(a-x)?y=f(x)的圖象關(guān)于x=a對(duì)稱(chēng).(2)y=f(x+a)是奇函數(shù)?f(a+x)=-f(a-x)?y=f(x)的圖象關(guān)于點(diǎn)(a,0)對(duì)稱(chēng).(3)若函數(shù)y=f(x)滿足f(a+x)=f(b-x),則y=f(x)的圖象關(guān)于直線x=a+b特別地,當(dāng)a=b時(shí),即f(a+x)=f(a-x)或f(x)=f(2a-x)時(shí),則y=f(x)的圖象關(guān)于直線x=a對(duì)稱(chēng).(4)若函數(shù)y=f(x)滿足f(x)+f(2a-x)=2b,則y=f(x)的圖象關(guān)于點(diǎn)(a,b)對(duì)稱(chēng).特別地,當(dāng)b=0時(shí),即f(a+x)+f(a-x)=0或f(x)+f(2a-x)=0時(shí),則y=f(x)的圖象關(guān)于點(diǎn)(a,0)對(duì)稱(chēng).【教材回歸】概念思考辨析+教材經(jīng)典改編1.思考辨析(在括號(hào)內(nèi)打“√”或“×”)(1)函數(shù)y=f(x+1)是偶函數(shù),則函數(shù)y=f(x)的圖象關(guān)于直線x=1對(duì)稱(chēng).()(2)函數(shù)y=f(x-1)是奇函數(shù),則函數(shù)y=f(x)的圖象關(guān)于點(diǎn)(1,0)對(duì)稱(chēng).()(3)若函數(shù)f(x)滿足f(x-1)+f(x+1)=0,則f(x)的圖象關(guān)于y軸對(duì)稱(chēng).()(4)若函數(shù)f(x)滿足f(2+x)=f(2-x),則f(x)的圖象關(guān)于直線x=2對(duì)稱(chēng).()2.(人教A必修一P87T13改編)函數(shù)f(x)=x+1A.(0,0) B.(0,1) C.(1,0) D.(1,1)3.已知函數(shù)y=f(x+2)-3是奇函數(shù),且f(4)=2,則f(0)=.

4.若偶函數(shù)y=f(x)的圖象關(guān)于直線x=2對(duì)稱(chēng),且當(dāng)x∈[2,3]時(shí),f(x)=2x-1,則f(-1)=.

【考向核心題型】考點(diǎn)一函數(shù)的對(duì)稱(chēng)性【典例】1.(2023·全國(guó)乙卷節(jié)選)已知函數(shù)f(x)=1x+aln(1+x),是否存在a,b,使得曲線y=f1x關(guān)于直線x=b對(duì)稱(chēng)?若存在,求a,【思維建?！?.函數(shù)y=f(x)關(guān)于直線x=a對(duì)稱(chēng)?f(a+x)=f(a-x),或f(2a+x)=f(-x).2.函數(shù)y=f(x)關(guān)于點(diǎn)(a,b)對(duì)稱(chēng)?f(2a-x)+f(x)=2b或f(a-x)+f(a+x)=2b.3.函數(shù)y=f(a+x)的圖象與函數(shù)y=f(b-x)的圖象關(guān)于直線x=b?a【變式訓(xùn)練】1(2024·新高考Ⅰ卷節(jié)選)已知函數(shù)f(x)=lnx2?x+ax+b(x-1)證明:曲線y=f(x)是中心對(duì)稱(chēng)圖形.考點(diǎn)二對(duì)稱(chēng)性與周期性【典例】2.(2025·?？谡{(diào)研)已知函數(shù)f(x)的定義域?yàn)镽,fx+12為偶函數(shù),f(2-x)+f(x)=0,f13=-12A.12 B.1C.0 D.-1【典例】3.(多選)(2025·安徽名校聯(lián)考)已知函數(shù)f(x),g(x)的定義域均為R,其中f(x)的圖象關(guān)于點(diǎn)(1,1)中心對(duì)稱(chēng),g(x)的圖象關(guān)于直線x=2對(duì)稱(chēng),f(x)-g(2+x)=4,g(2)=3,則()A.f(-x)+f(x)=0 B.f(2026)=-5C.g(2026)=-1 D.2026∑k=1f(【變式訓(xùn)練】2.已知函數(shù)f(x)的定義域?yàn)镽,f(x)=g(x-1)+2,若函數(shù)g(x)為奇函數(shù),g(x+1)為偶函數(shù),且f(2)=1,則23∑k=1gA.-1 B.0 C.1 D.2【變式訓(xùn)練】3.(多選)(2025·保定質(zhì)檢)已知f(x+1)是奇函數(shù),f(x)的圖象關(guān)于直線x=-1對(duì)稱(chēng),則下列結(jié)論正確的是()A.f(x)是周期為4的周期函數(shù)B.f(x-5)為偶函數(shù)C.f(x)的圖象關(guān)于點(diǎn)(-3,0)對(duì)稱(chēng)D.f(5)=0考點(diǎn)三對(duì)稱(chēng)性、周期性與單調(diào)性【典例】4.(多選)(2025·杭州調(diào)考)已知定義域?yàn)镽的函數(shù)f(x)在(-1,0]上單調(diào)遞增,f(1+x)=f(1-x),且圖象關(guān)于(2,0)對(duì)稱(chēng),則()A.f(0)=f(-2)B.f(x)的周期T=2C.f(x)在(2,3)上單調(diào)遞減D.f(x)滿足f(2025)>f(2026)>f(2027)【思維建?！拷鉀Q函數(shù)性質(zhì)的綜合問(wèn)題,一般要利用周期性與對(duì)稱(chēng)性縮小自變量的值或轉(zhuǎn)換自變量所在的區(qū)間,然后利用單調(diào)性比較大小或解不等式.【變式訓(xùn)練】4.(多選)(2025·齊魯名校聯(lián)盟聯(lián)考)已知函數(shù)f(x)的定義域?yàn)镽,f(2x+1)為奇函數(shù),f(4-x)=f(x),f(0)=2,且f(x)在[0,2]上單調(diào)遞減,則()A.f(1)=0B.f(8)=2C.f(x)在[6,8]上單調(diào)遞減D.f(x)在[0,100]上有50個(gè)零點(diǎn)【知識(shí)拓展】抽象函數(shù)1.我們把不給出具體解析式,只給出函數(shù)的特殊條件或特征的函數(shù)稱(chēng)為抽象函數(shù),解決抽象函數(shù)問(wèn)題的兩種常用方法有:函數(shù)性質(zhì)法和特殊值法.2.常見(jiàn)的抽象函數(shù)模型(1)f(x+y)=f(x)+f(y)可看作f(x)=kx的抽象表達(dá)式;(2)f(x+y)=f(x)f(y)可看作f(x)=ax的抽象表達(dá)式(a>0,且a≠1);(3)f(xy)=f(x)+f(y)可看作f(x)=logax的抽象表達(dá)式(a>0,且a≠1);(4)f(xy)=f(x)f(y)可看作f(x)=xα的抽象表達(dá)式.一、抽象函數(shù)求值【典例】1.(2025·南京部分學(xué)校聯(lián)考)已知函數(shù)f(x),對(duì)任意x,y∈R,滿足f(x+y)f(x-y)=f2(x)-f2(y),且f(1)=2,f(2)=0,則f(1)+f(2)+…+f(90)的值為()A.-2 B.0 C.2 D.4二、抽象函數(shù)的性質(zhì)【典例】2.(多選)已知函數(shù)f(x)的定義域?yàn)閧x|x≠4k+2,k∈Z},且f(x+y)=f(x)+f(A.f(0)=0B.f(x)為偶函數(shù)C.f(x)為周期函數(shù),且4為f(x)的周期D.f(2027)=-1【變式訓(xùn)練】1.f(x)滿足對(duì)任意的實(shí)數(shù)a,b都有f(a+b)=f(a)·f(b),且f(1)=2,則f(2)f(1)+f(4)fA.2026 B.2024 C.1013 D.1012【變式訓(xùn)練】2.(2025·紹興質(zhì)檢)已知f(x)是定義在區(qū)間(0,+∞)上的增函數(shù),且fxy=f(x)-f(y),f(2)=1,如果x滿足f(x)-f1x?3≤2,則x的取值范圍為【限時(shí)訓(xùn)練】（限時(shí)：60分鐘）一、單選題1.(2025·聊城檢測(cè))函數(shù)y=2-x與y=-2x的圖象()A.關(guān)于x軸對(duì)稱(chēng)B.關(guān)于y軸對(duì)稱(chēng)C.關(guān)于原點(diǎn)對(duì)稱(chēng)D.關(guān)于直線y=x軸對(duì)稱(chēng)2.已知函數(shù)f(x)=2|x-a|的圖象關(guān)于直線x=2對(duì)稱(chēng),則a等于()A.1 B.2 C.0 D.-23.(2025·武漢模擬)已知函數(shù)y=f(x)的定義域?yàn)镽,且函數(shù)y=f(x+1)為偶函數(shù),函數(shù)y=f(x+2)-1為奇函數(shù),則()A.f?12=0 B.C.f12=0 D.f4.已知函數(shù)f(x)=x3+ax2+x+b的圖象關(guān)于點(diǎn)(1,0)對(duì)稱(chēng),則b=()A.-3 B.-1 C.1 D.35.已知f(x)的定義域?yàn)镽,f(x+1)為奇函數(shù),f(x+2)為偶函數(shù),當(dāng)x∈[1,2]時(shí),f(x)=-3x2+2,則f143A.-103 B.10C.-23 D.6.(2025·雅安診斷)已知函數(shù)f(x)的定義域?yàn)?-∞,+∞),?x∈R,f32+x=f12?x恒成立.當(dāng)x2>x1≥1時(shí),[f(x2)-f(x1)]·(x2-x1)>0,f(0)=-f(2),則不等式f(x)(xA.(-∞,0)∪(2,+∞) B.(0,2)C.(-∞,0)∪(1,2) D.(0,1)∪(2,+∞)7.(2025·廣州模擬)已知函數(shù)f(x)的定義域?yàn)镽,且f(x+1)+f(x-1)=2,f(x+2)為偶函數(shù).若f(0)=2,則115∑k=1A.116 B.115 C.114 D.1138.已知函數(shù)f(2x+1)是定義在R上的奇函數(shù),且f(2x+1)的一個(gè)周期為2,則()A.1為f(x)的周期B.f(x)的圖象關(guān)于點(diǎn)12C.f(2027)=0D.f(x)的圖象關(guān)于直線x=2對(duì)稱(chēng)二、多選題9.定義在R上的函數(shù)f(x),f(x+1)的圖象關(guān)于點(diǎn)(-1,0)對(duì)稱(chēng),恒有f(x-1)=f(3-x),且f(x)在[1,2]上單調(diào)遞減,則下列結(jié)論正確的是()A.直線x=1是f(x)的圖象的對(duì)稱(chēng)軸B.周期T=2C.函數(shù)f(x)在[4,5]上單調(diào)遞增D.f(5)=010.已知函數(shù)f(x)的定義域?yàn)镽,滿足f(x+3)+f(x+1)=0,且f(x+1)為偶函數(shù),則()A.f(2)=0 B.f(x)為偶函數(shù)C.f(x)為周期函數(shù) D.f(x+4)為偶函數(shù)11.函數(shù)f(x)與g(x)的定義域?yàn)镽,且f(x)g(x+2)=4,f(x)·g(-x)=4.若f(x)的圖象關(guān)于點(diǎn)(0,2)對(duì)稱(chēng),則()A.f(x)的圖象關(guān)于直線x=-1對(duì)稱(chēng)B.2024∑k=1f(kC.g(x)的一個(gè)周期為4D.g(x)的圖象關(guān)于點(diǎn)(0,2)對(duì)稱(chēng)三、填空題12.寫(xiě)出一個(gè)同時(shí)具有性質(zhì)①②③的函數(shù)f(x)=.

①f(x)是定義域?yàn)镽的奇函數(shù);②f(1+x)=f(1-x);③f(1)=2.13.已知函數(shù)y=f(x)-1是奇函數(shù),若曲線y=1+1x與曲線y=f(x)共有6個(gè)交點(diǎn),分別為(x1,y1),(x2,y2),…