第三单元基本初等函数（ I ）

1. (1) .R,R.(2): k;k>0, ;k

2. (1) ,(2): , , ;a>0,, ; , ;a0,x a ;=0,x ;

4. y=f(x)=a +k(a>0)[m,n].(1)h[m,n], =k, =max{f(m),f(n)}(2)h [m,n ],hn,f(x)[m,n], = , = .f(m)f(m)f(n)f(n)

1y=(m+2)x+2m-1,yx,m. k>0y=kx+b(k0),y(0,b). y=(m+2)x+2m-1,m+2>0. y=(m+2)x+2m-1yx,2m-1

y=kx+b(k0)k,k>0,;k0,x;b=0,;b

2f(x)g(x).g(x)f(x)f(x).(1)g(x)= ,f(x)(4,-7).(2)g(x)=-2 ,f(x)(-3,2). g(x)a. y=a (h,k),y=a +k.(1)f(x)g(x)= f(x)4-7f(x)= -7= -8x+9.(2)f(x)g(x)=-2 ,f(x)(-3,2)f(x)=-2 +2=-2 -12x-16.

1

y=a +bx+c(a0)abc.

(2):

:y=a +bx+c(abca0);

:y=a +k(ahka0);

:y= (a a0).

3

.

2.f(x)f(2)=-1,f(-1)=-1,f(x)8,. :.f(x)=a +bx+c(a0).,

y=-4 +4x+7.:.f(x)=a +n(a0).f(2)=f(-1),

x= m= .y=8,y=f(x)=a +8.f(2)=-1,a +8=-1a=-4.f(x)=-4 +8=-4 +4x+7.:.f(x)+1=0 =2, =-1,f(x)+1=a(x-2)(x+1)(a0),f(x)=a -ax-2a-1. =8, =8,a=-4a=0().f(x)=-4 +4x+7.

3y=-3 -6x+1. . y=-3 -6x+1=-3 +4 (-1,4);x=-1;(-,-1-1,+)4.,A(-

1,4)xB C yD(0,1)E(-2,1).

1.(2).

:f(x)=t, +bf(x)+c=0,

+bt+c=0.

7f(x)=| +2x|f(x)=t7

f(x)=tf(x)=|

+2x| f(x)=| +2x|

4 0,1) b

4f(x)=- +2ax+1-a0x12,a. ,x=a,f(x)[0,1].

y=a +bx+c(a>0)[m,n]: [m,n].(1) [m,n],f( )= ,f(m)f(n)(m,n) );(2) [m,n], n,f(x)[m,n],f(n),f(m).4. 2010)f(x)= -2ax+3 -1(a>0,0x1)f(x).

f(x)= -2ax+3 -1= +2 -1,

a>0a1f(x)0,1f(x)

f(0)=3 -1,f(1)=3 -2a;

0

512f(x)=m +(m-3)x+1x,m. ,. (1)m=0,f(x)=-3x+1,x ,,...2(2)m0,f(0)=1,(0,1).3m

m>0,f(x),2.12,

8 m1m9, 0

(1)f(x)=2 -3x-k,x= . f(x)=0-1x1,

k-1. f(x)=0-1x1,

-1k5.,k

2f(x)=2 -3x-kx. f(-1)0 0 f(1)0, f(3)0 2+3-k0 9+8k0 2-3-k0, 18-9-k05k9.--113k(59).

y= -2ax-1[0,2]. x=0, =-1; x=2, =4-4a-1=3-4a. y= -2ax-1x=a,a,,a[0,2],. a

10.f(x)=| -2ax+b|(xR)f(x)f(0)=f(2)f(x)x=1 -b0 f(x)a+f(x) -b. .:f(x)= x=aa0..

0t1g(t)=f(1)=1t1f(x)t,t+1,g(t)=f(t)= -2t+2.

12. f(x)= +ax+af(x)-x=0 .(1)a;(2)f(0)f(1)-f(0) .1g(x)=f(x)-x= +(a-1)x+a,

a(0, ).

(2)f(0)f(1)-f(0)=f(0)g(1)=2 ,h(a)=2 .

a>0h(a)

0

(1)f(x)-x=0 +(a-1)x+a=0. , ,

a (0, )

(2)

f(0)f(1)-f(0)=f(0)

f(0)f(1)-f(0)< .

1. (1): =aaa(na (nN*); = (a0); (a0,nN*).(2): (m,nZ);

(m,nZ);

(m,nZ,a0);

(nZ).1

2. , =ax n>1,nN*.n, n,, (a>0); (a>0,m,nN*,n>1); (a>0,m,nN*,n>1).3. a>0,b>0, (r,sQ); (r,sQ); (rQ).4. anay= (a>0,a1)

5. y>10

1.(1)

(2)

(3)a,b -6x+4=0,a>b>0, . . (1)=

(2)=

(3)a+b=6,ab=4,a>b>0, (1),,.(2),,;,.(3),..1. :(1)

(2)

(3) =3, .

(1)=

(2)=

(3) =3,

2y= ,(1);(2);(3).

,,,. (1)

:y= (x-2),:

y= (x

1,,:

(2)

3.(1)y= ; (2)y=

(3)y= y= (a>0,a1)R,y= f(x);,.:1yxb1>a>b.:B

(1)x0{xR|x0}x0 0y= 1.(0,1)(1,+).(2) +1>0,R.y= ,0<

Ay= ,1-xR,y= ;B,x=0, -1=0;C,y1;D[0,1). A 4(12)Rf(x)2,x(0,1),f(x)=(1)f(x)[-1,1];(2):f(x)(0,1). f(x)[-1,1],f(x)(-1,0),x=1,0;. (1)x(-1,0),-x(0,1).f(x),f(x)=-f(-x)= .2

f(0)=-f(0),f(1)=f(-2+1)=f(-1)=-f(1),f(0)=f(1)=f(-1)=0.4[-1,1], .6(2)x(0,1),f(x)= .0<

4. x -b=0(bR).1b2

a>0a1,f(x)= [-1,1]14,a. x=1,f(x), +2a-1=14, +2a-15=0,a=3(a=-5). (1)a>10

10. 2009f(x)= -x-a(a>0,a1)a.:f(x)y=axy=x+a.,a>1;00,f(x)+k(1-x)f(x)>0.

1f(x)=

f(x)=0x=1.x

12. Rf(x)= .(1)a,b;(2)tR,f( -2t)+f(2 -k)

f(x),f( -2t)+f(2 -k)0,=4+12k1,3 -2t-k>0tR,=4+12k

1. (1):, x , ,a,N.(2) , ;10, 1, (3): (4): ,N .(5): ,N .aNN>010lgNe=2.718 28lnN

2. a>0,a1,M>0,N>0,(1) ;

(2) ;

(3) .3. (1):(2)(a,b,c>0a,b,c1): , , ,

1-1

4. :, , , . (0,+)R5. R01 y>0 y0 0

6. y= (a>0,a1)y= (a>0,a1,x>0) , .y=x 1.(1)

(2)lgx+lgy=2lg(x-2y), . ,.

(1)= = =

(2)x>0,y>0,x>2y.lgx+lgy=2lg(x-2y), xy= , -5xy+4 =0, x=4y(x=y). =4, =4. (1),.(2)(2):x>2y>0, .

1. ,.(1) ;(2) a>0,a1, .: (1)= =

(2),

, xy=3, x=2y, ,

2a,b,c, .: . ,. =k(k>0,k1),

:(1); .(2): (a>0,a1,N>0,N1).2. x,y,zR+,(1)3x,4y,6z;(2): . (1) =k,k>1,

k>1,,4y-6z

(2)(1)

3 ( )

A. 0 B. 1 C. 2 D. 3 y= y= ,,. , ,. 1.

3. ( )A. 0 B. 1 C. 2 D. 3 (),. C400,m>n;2m>0,n>0, >1,m>n.

(1)a10a1 0.(2)11.

5(12) 1 2 3 ,.1,a- 0 aa1x1.(-,1).2 f(x)1(,1)...4

5. f(x)= (a0,a1).1f(x)2f(x). f(x)= , 0,(1+x)(1-x)0,x+1(x-1)0,-1x1,f(x)(-1,1).(2)f(x).f(-x)= = =-f(x)f(x).

,;

,

.

10. (2008)a>1,cx[a,2a]y[a, ] ,a . , ,y= .

,y= [a,2a],

x=a,y= ;x=2a,y=a.

a=2. {2}

11. f(x)= (a>0,a1)(1)f(x);(2)a>1f(x)>0x.

12. a,bR,a2,(-b,b)f(x)= .(1)b;(2)f(x).

(2) , (-b,b), ,

b(0, ),- -b<

1. : ,x , .2. (y=x,y= ,y= ,y= ,y= ).

3. (1) ,(2)>0,,[0,+) .(3)

1f(x)= ,m,f(x):(1)?(2)?(3)?(4)?(5)(4),(0,+)? (1)(2)(3)(4)m,(5). (1)f(x),

m=1.

(2)f(x), m=-1.

(3)f(x),

(4)f(x), ,m= .(5)(4)m=m= , ,f(x)= (0,+),;m= , ,f(x)= (0,+).m= ,,.

1.y= ,m( )A. -1m2 B. m=1C. m=2 D. m=1m=2: , ,m=1m=2., -m-20,-1m2.m=1m=2. D 2 f(x), g(x).(1)f(x),g(x);(2)x:f(x)>g(x)?f(x)=g(x)?f(x)

,f(x)g(x).

(1):y= ();;.(2)(2),.,x,g(x){x|x0},f(x)

3:(1) ;

(2) ;

(3) ,, (1)y= (0,+), , , .(2) >1,0<

,:(1),;(2),;(3),,.3. 0

4(12) (0,+) , f(x)= (pZ) ,xR,f(x)-f(-x)=0.(1)p,f(x);(2)(1)f(x),g(x)=-qf(x)+(2q-1) +1,q(q0,-1

(2)g(x)=-q +(2q-1) +1.t= ,G(t)=g( )=-q +(2q-1)t+1(t0).6t= (-,0),x(-,-4],t[16,+);x(-4,0),t(0,16).8G(t)[16,+),(0,16),g(x)(-,-4],(-4,0),G(t)t=16,10t= q= .

q,q= .12 .,,,;,,.

4. f(x)= (kZ)f(2)

y= (mN*)y,(0,+), a. (0,+),

10. :y= ;y= ;y= ;y= y= ;y= ;y= ,,

,y= .ACE,

11. (2010f(x)= f(4)=- .(1)m;(2)f(x)(0,+).

12. f(x)= (nZ)[0,+), . : >0,-13x3-3x3x3-3x

1. :y=f(x)(xD), j y=f(x)(xD).:y=f(x) ,f(x)=0xf(x)=0y=f(x)x

2. f(x)=0y=f(x) y=f(x) .x3. ::f(x)=0;:,y=f(x),.

4. ,y=f(x)[a,b], ,y=f(x)(a,b),c(a,b), ,cf(x)=0..f(a)f(b)0),a0) x

6. :[a,b]f(a)f(b)

1.(1)f(x)=4x-3;(2)f(x)=- +2x+3;(3)f(x)= -3x+2;(4)f(x)=x- +2. ,,. (1)4x-3=0,x= ,f(x)=4x-3 .(2)- +2x+3=0, -2x-3=0, =-1, =3,f(x)=- +2x+3-1,3.(3) -3x+2= +2 -2 -4x+x+2= (x+2)-2x(x+2)+(x+2)= (x+2)=0, =1, =-2.f(x)= -3x+21,-2,1.(4)x- +2= =0, =1, =-3,

f(x)=x- +21,-3.

,,.1. .(1)f(x)= -1; (2)f(x)= (1) -1=0,x=1,f(x)= -11.(2) , =-1,f(x)= -1,. 2f(x)= +2 -3x-6(0.1). ,m,n,f(m)f(n)

f(1)=-60,x(1,2),f(x)=0.,:

a=1,b=2f(1)=-6,f(2)=4(1,2)f(1.5)=-2.6250(1.5,1.75)

f(1.625)=-1.3027

|1.734375-1.71875|0.11.7343751.71875. ,,,;,(),.2. y= -x-1[1,1.5],,(0.1). f(1)=-10,y= -x-1,[1,1.5].[1,1.5] =1.25,f(1.25)-0.3

[1.25,1.5] =1.375,f(1.375)0.22>0,f(1.25)f(1.375)

(1)y=f(x),f(a)f(b)

,f(x)=x,,f(x)=x. (1)f(x)= -x-3, , ,2 =-1 =3,3-1f(x).4(2)f(x),f(x)=a +(b+1)x+(b-1)=x,a +bx+(b-1)=0.6 -4a(b-1)>0,bR,8 -4(4a)

4. f(x)=- +2ex+m-1,g(x)=x+ (x>0).(1)g(x)=m,m;(2)m,g(x)-f(x)=0. (1):g(x)=x=e,g(x)[2e,+).m2e,g(x)=m.:g(x)=x+ (x0).g(x)=m,m2e.:g(x)=m, -mx+ =0(x>0)., ,m2e.(2)g(x)-f(x)=0,g(x)f(x).g(x)=x+ (x>0).

f(x)=- +2ex+m-1=- +m-1+ ,x=e,,m-1+ ,m-1+ >2e,m>- +2e+1,g(x)f(x),g(x)-f(x)=0.m(- +2e+1,+). nn.. .

10. x+lg x=3, , .:3

12. (2008)f(x)= .(1)f(x)=2,x;(2) f(2t)+mf(t)0t[1,2],m. f(x)=

(1)f(x)=2, , =0, , >0,x=

(2) 0t[1,2], -1>0,m-(1+ ).t[1,2],1+ [5,17],-(1+ )[-17,-5],m-5.m[-5,+).

1. (1)(2) (3)(4) (x>0)();(5) (m>0,m1,N>0);(6) (n>0);(7).y=kx+b(k0)y=a +bx+c(a0)2. y= (a>1), (n>0),y= (a>1), , ,

1,AB,()()(:MNCD).:(1)2,AB?(2)B500,?(3),BA? ,,,. ,,,,,.M(60,98),N(500,230),C(500,168),MNCD.

(1)2116168.(2)x>500, ()B500,0.3.(3),0x60, < ;60

1. 0.120.200.04(30)2040010250,x(250x400)20x+1025010(x-250).0.080.08.f(x)=0.08(20x+10250)-0.0810(x-250)=0.8x+400.f(x)250,400,x=400f(x)720.400720.

1 x (2)140a280a ,y;a y.a )a.

220092008.a10.010.43/4xy.1yxx2140

f(x)=a +c

,

.

x=4f(4)= =1.35.

1.31(),

.

y=a +bx+c. ,,.

2. 10 mO).

m4 m5 m.

m,.

y ba.

a0.

,b=-2,c=0.

.

32008100,1.2%.(1)y()x();(2)10,?(3)120? ,,. (1)1y=100+1001.2%=100(1+1.2%),2y=100(1+1.2%)+100(1+1.2%)1.2%=100(1+1.2%)2,3y=100 ,xy=100 .(2)10,100 112.7().(3)x120,100 =120,x= 15.28(),x=16().2024,120.

,,y= (N,p,x),.3. ,500 g,10%.(1)t,;(2)(0.1). (1)500,1,=500(1-10%)=500 ,2,=500(1-10%)2=500 ,t,=500 .(2)500 =250, =0.5,,tlg0.9=lg0.5,t= 6.6(),6.6.

41214 m,126 11ma21m 31m,1m xmx14x14,:x ,xm, m. (1)xm(x14),x ,1(14-x) ,2 a.3

y== (0x14).4y =35a. ,x=12 m =35a.6

(2)x14, 14=72a, .8y= = (x14).9t=x+ ,+),y1=x+ 14,+).10x=14m = =35.5a.1112m. 12 .

4. 13,6.

,131.6, (). 131.6,:f(A,B,E)=0.55+0.4+0.9=1.85();f(B,D,E,F)=0.4+0.5+0.9+0.1=1.9().ABEBDEF

ABCDEF5264610.550.40.60.50.90.1

u(:km/h):md.l:m50 km/h.Q

=

u=50Qmax= .

u=50 km/hQ.

.

u=50Qmax= .

u=50 km/hQ.

10. 160..11296100.y()x .:26 240

11. 2008.()()354 000(1)(2)13(3)11.(=-100%)

12. (2009)m..256;x x.y.(1)yx;(2)m=640y?

(1)n(n+1)x=m,n= -1,

(2)1f(x)=

f(x)=0 =512,x=64.

0

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第三单元 基本初等函数（ I ）

第三单元基本初等函数（ I ）