2.8函數(shù)的零點(diǎn)與方程的根高考新風(fēng)向·回顧教材思維引導(dǎo)回歸本質(zhì)(2024新課標(biāo)Ⅱ,6,5分,中)設(shè)函數(shù)f(x)=a(x+1)2-1,g(x)=cosx+2ax.當(dāng)x∈(-1,1)時(shí),曲線y=f(x)與y=g(x)恰有一個(gè)交點(diǎn).則a= ()A.-1B.12創(chuàng)新點(diǎn)本題看似是方程的解(函數(shù)零點(diǎn))問(wèn)題,通過(guò)恒等變形分離變量得a=cosx+1x2+1,進(jìn)一步運(yùn)用函數(shù)知識(shí)求解答案D五年高考考點(diǎn)函數(shù)的零點(diǎn)1.(2024新課標(biāo)Ⅱ,8,5分,中)設(shè)函數(shù)f(x)=(x+a)·ln(x+b).若f(x)≥0,則a2+b2的最小值為()A.18答案C2.(2018課標(biāo)Ⅰ理,9,5分,中)已知函數(shù)f(x)=ex,x≤0,lnx,x>0,g(x)=f(x)+x+a.若g(xA.[-1,0)B.[0,+∞)C.[-1,+∞)D.[1,+∞)答案C3.(2017課標(biāo)Ⅲ,文12,理11,5分,中)已知函數(shù)f(x)=x2-2x+a(ex-1+e-x+1)有唯一零點(diǎn),則a=()A.-12答案C4.(2019天津文,8,5分,難)已知函數(shù)f(x)=2x,0≤x≤1,1x,x>1.若關(guān)于x的方程f(x)=-14x+a(a∈A.54B.5C.54,94∪D.54,94答案D5.(2021北京,15,5分,中)已知函數(shù)f(x)=|lgx|-kx-2,給出下列四個(gè)結(jié)論:①當(dāng)k=0時(shí),f(x)恰有2個(gè)零點(diǎn);②存在負(fù)數(shù)k,使得f(x)恰有1個(gè)零點(diǎn);③存在負(fù)數(shù)k,使得f(x)恰有3個(gè)零點(diǎn);④存在正數(shù)k,使得f(x)恰有3個(gè)零點(diǎn).其中所有正確結(jié)論的序號(hào)是.

答案①②④6.(2022天津,15,5分,難)設(shè)a∈R,對(duì)任意實(shí)數(shù)x,用f(x)表示|x|-2,x2-ax+3a-5中的較小者.若函數(shù)f(x)至少有3個(gè)零點(diǎn),則a的取值范圍為.

答案[10,+∞)三年模擬基礎(chǔ)強(qiáng)化練1.(2024北京延慶一中月考,5)已知函數(shù)f(x)=12x?x13A.0,C.12,1D.(1,答案B2.(2025屆江西南昌期中聯(lián)考,6)已知函數(shù)f(x)=|ln(x+1)|-k有兩個(gè)零點(diǎn)a,b(a<b),則a+2(b+1)的取值范圍是()A.[22-1,+∞)B.(2,+∞)C.(0,2)D.(0,22]答案B3.(2024山西長(zhǎng)治四中月考,7)定義運(yùn)算:x*y=xy,x>y,x2,x≤y,例如:2*3=22=4,3*2=3×2=6.若函數(shù)f(x)=x*(2-xA.(0,1)B.(1,+∞)C.(0,+∞)D.(-∞,0)答案A4.(2024廣東潮州第四次測(cè)試,7)已知函數(shù)f(x)=x2?8x+8,x≥0,2x+4,x<0.若互不相等的實(shí)數(shù)x1,x2,x3滿足f(x1)=f(x2)=f(x3),則xA.(2,8)B.(-8,4)C.(-6,0)D.(-6,8)答案A5.(2024安徽皖江名校聯(lián)盟二模,6)已知函數(shù)f(x)=ax?1?1a-2(a>0且a≠1)有兩個(gè)零點(diǎn),則實(shí)數(shù)aA.0,1C.(0,1)D.(1,+∞)答案A6.(多選)(2025屆重慶八中期中,9)已知函數(shù)f(x)=x2?kx+1,x≤0,log2x,x>0,下列關(guān)于函數(shù)yA.當(dāng)0<k<1時(shí),有3個(gè)零點(diǎn)B.當(dāng)k>1時(shí),有1個(gè)零點(diǎn)C.當(dāng)k<0時(shí),有8個(gè)零點(diǎn)D.當(dāng)k=-4時(shí),有8個(gè)零點(diǎn)答案BD7.(多選)(2025屆吉林長(zhǎng)春東北師大附中開(kāi)學(xué)考,9)已知函數(shù)f(x)=?x2?4x,x≤0,|log2x,x>0,若x1<x2<x3<x4,且f(x1)=f(x2)=A.x1+x2=-4B.x3x4=1C.1<x4<4D.0<x1x2x3x4≤2答案AB8.(2025屆上海同濟(jì)大學(xué)附中月考,12)對(duì)于定義在集合D上的函數(shù)f(x),若存在實(shí)數(shù)x0滿足f(x0)=x0,則把x0叫做f(x)的一個(gè)不動(dòng)點(diǎn),已知f(x)=x2+2mx+4,D=[1,2]沒(méi)有不動(dòng)點(diǎn),則實(shí)數(shù)m的取值范圍是.

答案(-∞,-2)∪?9.(2025屆重慶楊家坪中學(xué)期中,13)已知函數(shù)f(x)=|ln|x+2||-m,m為正常數(shù),則f(x)的零點(diǎn)之和為.

答案-8能力拔高練1.(2025屆重慶八中開(kāi)學(xué)考,6)定義在R上的函數(shù)f(x)滿足f(x)=2f(|x|)+x2-x,則函數(shù)g(x)=xf2(x)-1x的零點(diǎn)個(gè)數(shù)為()A.3B.4C.5D.6答案B2.(2025屆江蘇南京期中,8)已知f(x)=-x2+2|x|,若關(guān)于x的方程f2(x)+mf(x)+n=0(m,n∈R)恰好有三個(gè)互不相等的實(shí)根,則實(shí)數(shù)m的取值范圍為()A.(-∞,-1)B.(-∞,0]C.(-∞,-1)∪(0,+∞)D.(-∞,-1)∪{0}答案D3.(2025屆湖南衡陽(yáng)一中開(kāi)學(xué)考,8)已知函數(shù)f(x)=2x,x≤0,lnx,x>0,g(x)=|x(x-2)|,若方程f(g(x))+g(x)-A.(1,+∞)B.[1,+∞)C.(-∞,1)D.(-∞,1]答案C4.(多選)(2025屆河北名校聯(lián)盟聯(lián)考,10)已知函數(shù)f(x)=ex+2x-2,g(x)=2lnx+x-2的零點(diǎn)分別為x1,x2,則()A.2x1+x2=2B.x1x2=ex1+lnC.x1+x2>43D.2x1x2<e答案ACD5.(多選)(2025屆江蘇百校聯(lián)考,11)已知函數(shù)f(x)=|4x?1|,x<12,|log2x,x≥12,若存在實(shí)數(shù)m,使得方程f(x)=m有四個(gè)不同的實(shí)數(shù)解x1,x2,x3,A.f(x3x4)=0B.x1+x2<0C.x2+f(x3)>1D.x3+f(x2)>1答案ABD6.(2025屆湖北黃岡中學(xué)月考,14)設(shè)a、b分別是方程log2x+x+2=0與2x+x+2=0的根,則a+b=.

答案-27.(2025屆江西九校期中聯(lián)考,14)已知函數(shù)f(x)=x2+2x+3,x≤0,lnx,x>0,若存在實(shí)數(shù)x1,x2,x3且x1<x2<x3,使得f(x1)=f(x2)=f(x3)=a,則x3(答案e38.(2025屆江蘇“決勝新高考”聯(lián)考,14)設(shè)函數(shù)f(x)=|xe-x+1+1|,若關(guān)于x的方程f2(x)-52f(x)+m=0有5個(gè)不相等的實(shí)數(shù)根,則實(shí)數(shù)m的取值范圍是答案1,創(chuàng)新風(fēng)向練1.(創(chuàng)新知識(shí)交匯)(2025屆山東臨沂開(kāi)學(xué)考,8)已知函數(shù)f(x)=m2cosx-m·2x+1的圖象和函數(shù)g(x)=m2x?1-3的圖象有唯一交點(diǎn),則實(shí)數(shù)m的值為(A.1B.3C.-1或3D.1或3答案D2.(創(chuàng)新知識(shí)交匯)(20