U=1,2,3,4,5

A=1,2,4,B=2,3,4,5

?UA?B

?A∩?B≠?

U

U

A∪B=U

A∩B?A∪B

A=?2,?1,0,1,B=0,1,2,3

U=R

?2,?1

?2,1

?2,?1,1

?2,?1,0

x

x?a≤2

A=x0≤x≤4

B=x|m?3≤x≤m+3

a

①

②

m

①

②

?2,4?A∪B

A∩B=A

①A∩B=B②x∈A

x∈B

③B∩?RA=?

A={x∈R(x?1)(x+2)>0},B={x∈Ry=x+a,y∈R}

a=1

A∩?RB

a

A=x,x,x,xxi∈?1,1,i=1,2,3,4

1

2

3

4

a,a,a,a,b,b,b,b∈A

ab+ab+ab+ab=0

a,a,a,a

b,b,b,b

1234

1

2

3

4

1

2

3

4

1

1

2

2

3

3

4

4

1

2

3

4

B?A

B

B

x,y∈M

x+y∈M

x?y∈M

M={x|x=3k,k∈Z}

M1M2

M1∪M2

B

A

n

m

=xx=,m∈A,n∈B

A=2,4

k

2

B

A

B=xx=?1,k∈A

∪B

k

A

N*

Pk

A

x

y

x?y≠k

B=1,4,5,8,11

A=a,a,a,a?1,2,3,4,5,6

A

P2

A

P3

1

2

3

4

A?1,2,?,11

P4

P7

A

Ω={xx?U}

A?Ω

①

M,N∈A

M∪N∈A

②

M∈A

?UM∈A

U={1,2,3}

M,N∈A

?∈A

M∩N∈A

3

m

1

n

2

2

m>0n>0m+4mn+3n=m+n

+

4+23

3+22

x,y

2x+y=2

1

4

xy

y+x2

1

2x

1

y

x

4+2y

+

11

M=max4x+,+y

yx

maxa,b

a

b

xy

M

22

3

8

9

3

a

3

b

a>0b>0

x>0y>0

+2b=2

2a+

2x+y=1

1

x

2

y

+

b

a

b

1?a

0<a<1

+

≥1

b

1

4

1

2

1

2

,+∞

,+∞

0,

0,4

2x+y?2xy=0

x>0,y>0

2x+y>k2+k?8

k

(?4,3)

[?4,3]

(?3,4)

y

[?3,4]

x

x?1y?2=2

3x+2y>m

m

?∞,4+62

?∞,7+43

6+42,+∞

8+43,+∞

a

b

2

a

a>0,b>0

2a+b=2

t2?3t≤

+

2

a+bc+d≥ac+bd

1

x

1

y

2m+2

m

x+y=1

+

≥

a?2x2?2a?2x?4<0

x∈R

a

?∞,?2∪2,+∞

?2,2

?∞,?2∪2,+∞

?2,2

a>0,b∈R

ax?2x2+bx?6≥0

0,+∞

4a?b

22

32

?x∈R,mx2+2m?3x+4≤0

?∞,0

m

1,9

?∞,1∪9,+∞

?∞,1∪9,+∞

?x2+2x+m≤0

x∈[0,2]

x

2x?1>Kx2?1

K

x∈R

K

x∈1,+∞

K∈?2,2

x

R

fx

?x,y∈Rfx?fy=fx?y+2x?yy

f6=0

f0=1

f3=9

?x∈Rfx+f?x≥0

fx

f(x)

R

f(1)=?2

f(x)?f(y)=f(x+y)+f(x?y)

f(0)=0

f(x)

f(2)=1

f(x)

y=fx

?∞,0∪0,+∞

fxy=fx+fy?1

fx

1

f2=

x>1

f1024

2

fx<1

f2x+1>1

fx

x,y∈R

fx+y=fx+fy?2

f2=4

f1

gx=fx?2

gx

f?2025+f?2024+?+f?1+f0+f1+?+f2024+f2025

R

f(x)

?x,y∈R

f(x+y)=f(x)+f(y)+1

f(1)=1f(x)

R

f(0)

f(x)+1

?x∈[?1,1]

f(x)>m2?m?2

m

f(2x2?8x+4)+3f(x)>0

fx1?fx

2

fx

?x,x∈R

x1≠x2

>2

1

2

x1?x2

f2=12

?2,2

?∞,?2∪2,+∞

fm2≥2m2+8

m

?2,2

?∞,?2∪2,+∞

fx

R

fx+4=2fx

x∈0,4

fx=2x2?8x

3

4

?x∈?∞,t

fx≥?

t

?∞,?11

?∞,?5

?∞,?7

?∞,?3

2

3

fx

?∞,0∪0,+∞

f3=

xxfx?fx

xfx?2

x

?x1x2∈0,+∞

x1<x2

1

2

1

2

<2

x

>0

x2?x1

?∞,?3∪0,3

?3,3

?3,0∪3,+∞

0,3

fx=x2+5x+8gx=mx+3?5m

fx1=gx2m

x1∈?4,2

x2∈2,6

y=fx

?2,?6

f0=?2

fx?2

fx

x

gx=

y=fx

?x∈1,2,?x∈1,2

gx1≥mx2+4

m

1

2

fx

fx?1

1,0

fx+2

fx

0,2

f10<f19<f13

f13<f10<f19

f10<f13<f19

f13<f19<f10

fx

R

fx+1

fx+2

x1x2∈1,2x1≠x2

fx

x1?x2fx1?fx2>0

f2025=0

21

9

fx

?1,0

f?<f

5

8

x

x

x

x?x

x

x

fx=x

1

fx+=fx

2

gx=xx

hx=2xx?x?1

?1

(?2,3?a)

ax+b

4?x2

f(x)=

f(x)

f(x)

f(2t+1)+f(t?2)>0

x

fx=

x+1

fx

fx

0,+∞

x

fax2+3ax+f1?ax>0

x

a

fx=2x2?2x+1

fx

?∞,1

1,+∞

1,+∞

0,1

?∞,1

?∞,1

?∞,1

0,1

x

a?1

a+1

f(x)=

(a>0a≠1)

x

①

②

③

④

f(x)

f(x)

f(x)

f(x)

(0,1)

(0,0)

1

2

3

fx=a?

4

1

2+1

x

a

fx

x∈R

fmx2+f3?2mx>0

m

?ex+a

ex+1

R

f(x)=

a

y=f(x)

x

x

x

f(m?3)+f(3?9?2)>0

x≥0

m

x

x

fx=9?m?3?1

f2=?1

m=1

m

fx

?2,1

x+1

gx=2

x1∈?2,1

x2∈R

fx1≥gx2

m

f(x)=lnx?2?lnx

f(x)

f(x)

R

f(x)

(?∞,0)

3

1

(1,0)

4,5

f(log23)>f(?)

2

2

fx=logx?ax+6(1,2)

a

2

4,+∞

?∞,7

4,7

2

fx=2x?1?logx?2x+4

a=flog23b=flog34

1

2

c=flog45

abc

a>b>c

c>b>a

b>a>c

b>c>a

1+ax

f(x)=log2x?1

f(x)

5

x∈[,3]

f(x)<2m?1

3

fx=log1+2x?log1?2x(a>0,a≠1)

a

a

fx

fx

x

fx<0

fx=2025x+log2025x2+1+x?2025?x+1

fx2?2x+f(3x)>2

0,+∞

?1,0

?∞,?1

R

?∞,?1∪0,+∞

fx1?fx

2

fx

x1,x2

x1<x2

>?1

x1?x2

x

x

f1=1

1,+∞

?1,0∪0,3

flog3?1<2?log3?1

2

2

?∞,1

?∞,0∪0,1

2

fx=logx?ax+1

2

a=2

fx

fx

2,+∞

a

gx=4x?2x+1

x1∈0,1

x2∈?1,1

fx1≥gx2

a

3x+1

3x+a

fx=

a

x

x

gx=log?log+m

3339

x1∈3,27

x2∈0,2

gx1=fx2

m

1?ax

f(x)=log2x?1

f(x)(1,+∞)

x

?x

x

?x

f2+1+f2+1>logm2?2

x∈(0,+∞)

2

2x?2,0<x≤1

f(x)=1

2

g(x)=x2f(x)?1

f(x?1),x>1

log2x,x≤2

x2?8x+14,x>2

fx=

fx=k

x1

x2x3xx<x<x<x

4

x1+x2x3+x4

4

1

2

3

0,16

(18,+∞)

16,20

5?2x?2,x≥0

[16,+∞)

fx=

gx=f2x?m+2fx+2m

2x+3

x+1

,x<0

1,2∪3,4

1,2∪3,4

2,3∪4,+∞

log2x,x>0

4,+∞

f(x)=

f(x)=a

x,x,x,x

1234

1

x2+x+2,x≤0

4

x+xx

3

x<x<x<x

x4?

1

2

1

2

3

4

4

16

x+,x≥3

x

f(x)=

1

,x<3

x?3

f(x)(?∞,3)

F(x)=f(x)2?4af(x)+3a2

a

y=fx

x0fx0+f?x0=0

x0,fx0

x2+3x,x<0

kx+5,x≥0

y=fx

3?25,0

?∞,3+25

fx=

fx

k

0,3+25

?∞,3?25

D

fx

k

x∈D

fx?f?x≤k

fx

P0

P1

Pk

①

x

②fx=

1+x2

③

④

fx=x2+x+1

k

fx

Pk

a

1+2

a>0

fx=

Pk

a∈0,k

x

①

②

③

④

f(x)

x1≠x2

x

f(x)+f(?x)=0

f(x)?f(x)

x1,x2

1

2

<0

1

f(x)

x1?x2

①f(x)=?x②f(x)=x2③f(x)=|x|④f(x)=

x

y=fx

x1

x2

fx1fx2=1

gx=x

fx=2x?2

m,nn>m>0

mn

3

2

3

2

3

2

hx=x?a2a<

hx≥?t2+s?tx

,3

x∈,3

t∈R

s

A=yy=fx,x∈DB=yy=2m?x,x∈D

m∈R

B?A

y=fx

1,2

0,a

pm

y=3x

y=3x?1

p1

p1

fx=?x2+2tx+3

0,2

pm

π

6

π

4

x=

fx=cosωx?

ω>0

ππ

?,

312

fx

ω

3

2

5

3

15

2

π

3

π

2

f(x)=sin(ωx+)ω>0

T>

f(x)

ππ

(,)

63

2π

3

x=

ω

1

4

21

4

π

6

fx=3sinωx+ω>0

0,2π

ω

1913

66

138

63

1913

66

138

63

,

,

,

,

π

2

π

8

fx=sinωx+φω>0φ≤

x=?

fx

π

8

ππ

189

x=

y=fx

fx

,

ω

π

3

P3,1

φ

fx=sinωx?φω>0

0,

2π

3

,π

ω

2

3

5

3

0,

1,

58

23

1117

43

,

,

π

f(x)=2cos(ωx+)+2(ω>0)

ππ

[?,]

63

6

[0,π]

2

3

1

2

11

12

13

12

fx=sinxgx=cosxhx=fx+gx

y=fxgx

hx0,2π

hx

0,2π

y=fgx

π

3

fx=2sinωx+φ

ω

ω>00<φ<π

fx≤f

π

2

fx

0,

3

fx=22cosωx+φ(ω>0,0<φ<π)

fx

fx

π

2

0,

π

4

gx=fx+2

,m

m

π

f(x)=2cos(ωx+)(ω>0)

[0,2π]

6

ω

ππ

[?,]

24

ω∈N?

ω∈N?

|f(x)?m|<3

y=f(x)+t

ππ

[?,]

42

π

2

m?1sin2x?msinx+m?1>0

0,

1,+∞

?∞,4

2,+∞

4,+∞

2,4

3,4

4

2

cosx?tcosx+3>0

?∞,?4

?4,+∞

π5π

cos2x+2sinx?1?m≤0

?,

36

π

fx=23sin(2x?)+2

6

fx

fx

ππ

?,

64

ππ

124

f2x?2mfx+3m≥0

x∈?

,

fx=cos2x?2asinx+a+3a∈R

fx

R

π

2

fx>0

0,

a

fx=5

0,2π

a

π

6

π

6

π

2

fx=2sinx+

gx=2sinωx+

x1∈0,

π

2

x2∈0,

fx1=gx2+2

ω

π

6

1

ω

fx=2sinx+

ω>0

π

2

π

2

gx

x1∈0,

x2∈0,

fx1=gx2+2

ω

π

6

π

2

fx=asin2x?+ba>0,b∈R

0,

0,3

fx

x1∈0,

π

6

π

2

x2∈,m

fx1≥fx2

m

π

2

5π

12

fx=sin2x+φ0<φ<

f

=0

fx

ππ

?,

43

ω>0

gx=fx+

fωx

ω

ππ

π

12

x

2

x

π

3

4

+af?af

+

?2a+

,

a

24

123

π

3

fx=sin2x+,gx=2sin2x+acosx+1a∈R

y=fx

gx

ha

x2∈R

ha

fx1+1≥gx2

x1∈R

a

ππ

α∈?,

22

sinα=x

α=arcsinx

2y2?1=

α∈0,π

cosα=x

2

2

α=arccosx

2a

y∈0,1

(arccosy)?(arcsiny)=a

2a

π

cos

?sin

?sin

π

4a

π

4a