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第 1 节 函数与反函数

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第 1 节 函数与反函数. 第二章 函数. 要点 · 疑点 · 考点. 1. 映射 设 A , B 是两个集合,如果按照某种对应法则 f ,对于集合 A 中的任何一个元素,在集合 B 中都有惟一的元素和它对应,那么这样的对应叫做集合 A 到集合 B 的映射,记作 f:A→B . 给定一个集合 A 到 B 的映射,且 a∈A,b∈B. 如果元素 a 和元素 b 对应,那么,我们把元素 b 叫做 元素 a 的象,元素 a 叫做元素 b 的原象 - PowerPoint PPT Presentation

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  • 1

  • 1.ABfABABf:AB .ABaA,bB.abbaabf:ABAB.ABBAB.

  • 2.(1)x,yxf,yyxy=f(x) (2). 3. .4.. 5.. y=f(x)AC.yxx=(y)yCx=(y)xA.x=(y)(yC)y=f(x)(xA).x=f-1(y)y=f-1(x)

  • (1)D(2)y=-log3(x+1)(x0)(3)[-1+

  • (4) B (5) C4.-2-1012f(x)f(2)=1,f(1)=2,f(0)=0,( ) (A)f(x) (B)f(x) (C)f(x) (D)f(x) 5.y=f(x)f-1(x)=2x+1f(1)( ) (A)0 (B)1 (C)-1 (D)4

  • f:ABfcard(A)=3,card(B)=2f:ABf? 1.A=a,b,B=0,1f:ABf.

  • y=f(x)y= f-1(x)(1)y=f(x)()(2)y=f(x)x=f-1(y);(3)xyy=f-(x)(4)y=f(x)2. (1) y=1/2ln(x-5)+1(x5) (2)y=x2+2x(x0)

  • f-1(a)f(x)f-1(x)f-1(a)f(x)f-1(x)f(x)=a.3.f(x)=2x/(1+2x)(xR)f-1(1/3)f(x)f-1(x)f(a)=b, f-1(b)=a.4.f(x)=ax+kA(13)y=f-1(x)B(20)f(x).

  • . 5.y=f(x)y=f-1(x).

  • .y=x.

  • 1. 2..

  • 2

  • 1..

    2.f[g(x)]f(x)

  • CA B7/2

  • 5.y=f(x)[-12]1

    3f(x)__________________

    6.yx.10008002000700.400( ) (A)820 (B)840 (C)860 (D)880 C

  • f[g(x)]g(x)f(x).(1)g(x)=tt(g(x)) (2)x=(t) (3)g(x)=tx=(t)f[g(x)] (4)xtx(t)

  • f(x-2)=f(-x-2).f(x)f(a+x)=f(a-x)f(x)x=a..y=f(x)P(ab)y=g(x).3.y=x2+xy=g(x)(-23)g(x).

  • .4.A300kmB75km/hABC2h100km/hBv(I)Ax(km)At(h) (II)(AB)v

  • 5.800800xx=-800

    15005%2500200010%320005000 15%910000 45%

  • (1)f(x)13f(x)(2)2002103000? (3)26.78( ) (A)800900 (B)9001200 (C)12001500 (D)15002800 ..

  • 1.f(1-cosx)=sin2xf(x)t=1-cosx0t2f(x)0t2.2.

  • 3

  • 1.x.(1)(2)(3)(4)1. 2..x. 3.f(x)Af[g(x)]u=g(x)uAg(x)Ax.

  • 4..5..6..

  • (1)(--1] (2) [5+ (3) C

  • DA

  • y=f[g(x)]f(x)g(x)f[g(x)] 1.f(x)[ab]a+b0f(x2)

  • (3)(4)x0x0x.

  • xRax2+bx+c0.a=0a0..

  • (1)()()(2)()(3)()(). 4.f(x)=x2-2ax(0x1)M(a)m(a)M(a)m(a).

  • 1..2.. 3.f(x)xf[g(x)]x.

  • 4

  • (1)f(x)xf(-x)=f(x)f(x). (2)f(x)xf(-x)=-f(x)f(x) f(x)f(x) 1.

  • yy 2. (2) 3. (1).f(-x)=f(x)f(-x)=-f(x). f(-x)=f(x)f(-x)f(x)=0f(x)/f(-x)=1

  • (3) ()()()() (ab)()(-b-a)()(ab)(-b-a).

  • 1.f(x)=ax2+bx+c(2a-3x1)a___b____c___2.f(x)(xR)3f(1)1f(2)=a( ) (A)a2 (B)a-2 (C)a1 (D)a-1 3.f(x)x0f(x)=2x-1/2x-1/4( ) (A)f(x)0 (B)f(x)0 (C)f(x)+f(-x)0 (D)f(x)+f(-x)0 {1}{0}RDB

  • DA

  • 1. f(-x)+f(x)=0

  • .

  • 2.(1)f(x) F(x)=[f(x)+f(-x)]/2;G(x)=[f(x)-f(-x)]/2; (2)y=2x..

  • .-xxf(x)g(x).3.f(x)g(x)f(x)-g(x)=(1/2)xf(1)g(0)g(-2).

  • ..

  • 1

  • 5

  • 1. f(x) I I x1 , x2x1x2f(x1)f(x2)f(x).Ix1 , x2x1x2f(x1)f(x2)f(x)...y=x2x[0+]x(-0).

  • 2. y=f(x)y=f(x)()y=f(x).. 3.f(x)M(1)x1,x2Mx1x2(2)f(x1)-f(x2)(3)(4).

  • 4. f[g(x)]u=g(x)y=f(u)

    u=g(x) y=f(u) y=f[g(x)]

  • 1.(-0)( ) (A)f(x)=x2-4x+8 (B)g(x)=ax+3(a0)(C)h(x)=-2/(x+1) (D)s(x)=log(1/2)(-x)2.(-+)f(x)g(x)[0,+)f(x)ab0 f(b)-f(-a)g(a)-g(-b); f(b)-f(-a)g(a)-g(-b);f(a)-f(-b)g(b)-g(-a); f(a)-f(-b)g(b)-g(-a)( ) (A) (B) (C) (D) DB

  • (3) B (4) (--1)(-1+) (-11] (5) C

  • 1.f(x)=x+a/x(a0)..2.y=f(x)(0+)f(x)0F(x)=1/f(x)(-0)? (0+)x1x2.-x1-x2(-0).

  • ..

  • ..4.af(x)=loga(ax2-x)[24]?

  • f(x+g)=f(x)+f(y)f(x)f(y)=f(x+g)f(xy)=f(x)+f(y). .

  • (1).. (2).

  • 6

  • 1. y=f(x)xy(xy)y=f(x)(xy)y=f(x)y=f(x)xy(xy) 2.x,y.

  • (1)y=f(x)y=f(x+a)+b

  • (2)y=f(x)y=Af(x)(A0A101)

  • (3) y=f(x)y=f(-x)y y=f(x)y= - f(x)x y=f(x)y=-f(-x) y=f(x)y=f -1(x)y=x y=f(x)yy.yy=f(|x|) y=f(x)xxy=f(|x|)

  • 1.y=log2(x-1)y=2x___________________ ___________________ ______2.y=log(1/2)xxCC1CC2C1y=xC2________________3.y=f(|x|)y=f(x)( ) y y=x .y=-1-2xB

  • BA

  • f(x)=ax3+bx2+cx+d(a0).

  • 2.(1)y=2-2x;(2)y=log(1/3)[3(x+2)];(3)y=|log(1/2)(-x)|

    ()..

  • (1)(2)... 3.(1)0a1a|x|=|logax|( ) (A)1 (B)2 (C)3 (D)123

  • f(a)g(a)f2(a)g2(a)(2) (AA+CC)/2BB(AA+CC)/2AACCBB(AA+CC)/2 BB.

  • .

  • 2..3..

  • 7

  • 1. f(x)=ax2+bx+c(a0) f(x)=a(x-k)2+m(a0) f(x)=a(x-x1)(x-x2)(a0)

  • 3.f(x)=ax2+bx+c(a0)[pq]-b/2app-b/2aq-b/2aq. 4.f(x)=ax2+bx+c=0 x=-b/2a ax2+bx+c=0f(x)=ax2+bx+c(a0)

  • (1) 6 (2)19 (3)C 1.f(x)f(3+x)=f(3-x)f(x)=0x1,x2x1+x2_________.

    2.f(x)=2x2-mx+3x(-,-1]x(-1,+)f(2)= _______.

    3.xx2+(a2-1)x+(a-2)=011( ) (A)-1a1 (B)a-2a1(C)-2a1 (D)a-1a2

  • C

  • xRy=ax2+bx+c(a0)x

  • 2f(x)=mx2+(m-3)x+1xm

  • (1)(2) 3.f(x)=x2-4x-4[tt+1](tR)g(t).(1)g(t)(2)g(t)g(t)

  • .(). 4.f(x)=ax2+bx+cg(x)=-bxa,b,cabca+b+c=0(a,b,cRa0)(1)AB (2)ABxA1B1

  • f(x)=a(x-x1)(x-x2)x5.f(x)=ax2+bx+c(a0)f(x)-x=00x1x21/ax(x1x2)x1f(x)x2.

  • 2..1..

  • 8

  • 1. (1)aman=am+n (m,nZ)(2)aman=am-n (a0,m,nZ) (3)(am)n=amn (m,nZ) (4)(ab)n=anbn (nZ) 2. na(n1nN*)anxn=axann1nN*nana

  • 4. 5. (1)aras=ar+s (a0r,sQ) (2)aras=ar-s (a0r,sQ) (3)(ar)s=ars (a0r,sQ) (4)(ab)r=arbr (a0b0rQ)

  • 6. y=ax(a0a1)xR7.()

  • 8. a(a0a1)bNab=NbaNlogaN=b,aNlogaN log10NNlgN e=2.71828NlogeNlnN. 10. (1) (2)1loga1=0 (3)1logaa=1

  • 12.. y=logax(a0a1)(0+)(-+).y=logaxy=axy=logaxy=axy=x.

  • 13. y=logaxa10a1.y=axy=axy=xy=logax

    a>10

  • 1. (1/21) 2.1 3.D

  • 4.loga2logb20( ) (A)0ab1 (B)0ba1 (C)1ba (D)0b1a

    5.loga(x+1)+x22(0a1)( ) (A)0 (B)1 (C)2 (D) BC

  • (2)1x1x.

  • 2.f(x)lg(1-x)g(x)lg(1+x)f(x)g(x)| f(x) || g(x) |. .ka. 3.f(x)log2(ax-2xk)(a2k).

  • (1)y(2)a(3)logax. 4.yloga(a2x)loga2(ax)x(24)y[-1/80]a.

  • ...

  • 2..ylog2(x2-2x)yx2-2x.1..

  • 9

  • 1. ...

  • 2. ().. .().f(x)0yf(x)f(x)0(f(x)0)yf(x).

  • 3. .

  • .. yax+bx.

  • 2500m2C

  • CD

  • xN*. 1.()()23()?

  • 3.f(x)Df(x)f(x)D a,bDf(x)a,ba,bf(x)(x D). (1)y=-x3(2)a,b (2)y=2x-lgxa,b (3)y=k+ k. .

  • ??() 4.(120)36060.xyz()yzxssxs=-x+1080sx30.

    1/21/31/4()432

  • (2).

  • 2..1..