1. x
(Function)
F(x)

2. :A B f
A B
f xA
y,z  B
(x,y)  f (x,z)  f y=z
A = {1,2,3,4} B = {a,b,c}
f1 = {(1,a),(2,b),(3,a)} f2 = {(1,a),(2,b),(4,c),(1,b)}
1 1
a a
2 2
b b
3 3 c
c
4 4

3. Df = { x | y (x,y)  f
}
Rf = { y | x (x,y)  f
}
f (x,y)  f
y
x
y f
x (1,2)  f  f(1) = 2
f(x)
y = f(x)
f(x) =

4. (x,y1)  f (x,y2)  f
y1 = y2
(x,y1)  f (x,y2)
f
y1  y2

5. f={(x,y)RR|y = x2+1}
(x,y1)f (x,y2)f
y1  y2
x2+1  x2+1
x2  x 2
y1 = y2

6. f={(x,y)RR|x2+y2 = 1}
f
(0,1)f ( 02+12= 1)
(0,-1)f (02+(-1)2 = 1)
1  -1

7. Rf  B
f A B (f is function from A to
B)
f:AB
Rf = B f A
B
(f is function from A
onto B)
A = {1,2,3} f BA { a,b,c}
: = B
f1= {(1,a),(2,a),(3,b)} f1 : A  B
f2 = {(1,a),(2,b),(3,c)} f2 : A  B
f3 = {(1,b),(2,c),(3,a)} f3 : A  B

8. f:AB
yB , x  A f(x)
=y
B = Rf
y B
y = f(x)
x
x A
x A

9. f:QQ f(x) = 3x+4
yQ y=
f(x)
y = 3x+4
x = (y-4)/3  Q
f:NN f(x) = 3x+4
yN y = f(x)
y = 3x+4
X = (y-4)/3  N y=5 x=
1/3

10. f:RR f(x) = x2
yR y = f(x)
y = x2
x = y R y<0
f
yB xA f(x) y
y f(x) y
xA
y = -1 xA
x2  -1

11. :f:AB f
(one-to-one)
x1,x2  A y
B
(x1,y)  f (x2,y)  f x1 = x 2
1-1
f:A  B
x1  x2  f(x1)  f(x2)
f(x1) = f(x2)  x1 = x2

12. Direct method Contradiction method
Contrapositive method
f(x1) =
f(x2)
f(x1) = f(x2) x1
x2
x1 x2
x1 = x 2
x1 x2
f(x1)  f(x2)

13. f(x) = 2x+1
x
f(x1) = f(x2)
2x1+1 = 2x2+1
2x1 = 2x2
x1 = x 2

14. x1 x2
f(x1) = f(x2) x1 f(x1) = f(x2)
x2 2x1+1 = 2x2+1
2x1+1 = 2x2+1 2x1 = 2x2
2x1 = 2x2 x1 = x 2
x1 = x 2 f(x1)  f(x2)

15. x1,x2  Df x1 
x2
f(x1) = f(x2)
f(x) = x2
x1 = 2 x2 = -2
f(x1) = f(2) = 4
f(x2) = f(-2)= 4 f(x1)
= f(x2)
(x1)2 = (x2)2
x1 =  x2

16. : f:AB
f-1 f-1
f
f-1 : f(A)  A
A = {1,2,3} B = {a,b,c}
f = {(1,b),(2,a),(3,c)}
f-1 = {(a,2),(b,1),c,3)}
f f-1
a a
1 1
b b
2 2
c c
3 3

17. A = {1,2,3} B = {a,b,c}
f = {(1,b),(2,b),(3,c)}
f-1 = {(b,1),(b,2),c,3)}
f f-1
a a
1 b 1
b 2
2 c c
3 3
f-1
f

18. : f:AB
f f
f f
f
( f )
x1,x2  A yB
(x1,y),(x2,y)  f ( f(x1)=f(x2)=
y )
(y,x1),(y,x2)  f-1
f -1 ( f
)

19. f f
f
y B x1,x2 A
(y,x1),(y,x2)  f-1
(x1,y),(x2,y)  f
x1 = x 2
f-1

20. 1-1 1-1
: f:AB f :BA
-1
1-1
f:AB f f-1:
B A
f-1
f-1(y1) = f-1(y2)
x x = f-1(y1) = f-1(y2)
(y1,x) f-1 (y2,x)f-1
(x,y1)f (x,y2)f
y1 = y 2 (f
)
f-1

21. f-1
( Rf-1 = A)
xA
yB f(x) = y
(x,y)  f
(y,x)  f-1
x  Rf-1
A  f-1
Rf-1  A
Rf-1 = A
f-1

22. : A B
A B
B
AB
: f:AB g:BC
f g
gof
A C
(gof)(x) = g(f(x)) xA
gof = {(x,z) | yB (x,y)f (y,z)
g}

23. : f1-1A  B
: 1-1 :
g BC 1-1 f
go :
A C
1-1
1-1 1-1
f:AB g:B
C
gof(x1) = gof(x2)
g(f(x1)) = g(f(x2))
f(x1) = f(x2)
x 1 = x2

24. : f:AB g:BC gof : A
C
f:AB g:BC
zC ( C
Rgof )
Rg = C
z  Rg
y B (y,z)  g
Rf = B
yRf
xA (x,y)  f
(x,z)  gof
z  Rgof C  Rgof
R o C

25. : f:AB kN
f k 1 (k-1)
y B x k
f(x) = y
0 a 0 a
1 b 1 b
2 c 2 c
3 d 3 d
2- 3-
1 1

26. : f:AB k-1
n(B)
A n(A) 
k.n(B)
f:AB k-1 n(B)
x k
yB
A
k.n(B)
A n(A)  k.n(B)
x1
.
y1
xk y2
.
.
ym

27. (Pigeon-Hole Principle )
n
m n>m
1
2 1
2
3
4
J.P.G.L.
Dirichlet

28. x (floor of x)
x
x
4= 4
4.0000001= 4
4.9999999= 4
x(ceiling of x)
x
x
4 = 4
4.000000001 = 5
4.999999999 = 5

29. (Generalized Pigeon-Hole
Principle)
: f:AB n(A) = m n(B) = m
k
 
yB xA 
k 
f:AB n(A) = m n(B) = k m 
yB  k  1
xA
 
m 
 k  1
 
A k(
m  m
)  k  1
  k
m 
m  n( ) < k((  k   1m
 
+1) -1) =
m 
k 
 
y B xA

30.  89 
 12 
 
=
= 7.4167
=8

31. 1.
n2 ; n 0
1.1 f : Z  Z f(n) =
-n2 ; n < 0
n+1 ; n
1.2 f : Z  Z f(n) =
n3 ; n
x+1 ;xQ
1.3 f : R  R f(x) =
2x ;xQ
3x+2 ; x  Q
1.4 f : R  R f(x) =
3

32. x 1
1.5 f : R  R f(x) = ;x0
x
x
1.6 f : R  R f(x) = 2
x 1
3x - 1
1.7 f : R  R f(x) = ;x0
x
x 1
1.8 f : R  R f(x) =x  1 ;x1
2. f:RR f(x) = 2x
3
2 x  2x
g:RR g(x)=
2
f=g x 1

33. 3. f:RR f
3.1 f(x) = 6x-9
3.2 f(x) = 3x2-3x+1
3.3 f(x) = sin x
3.4 f(x) = 2x3-4
3.5 f(x) = 3x-2x
2
x 1
3.6 f(x) =
4. f g Z+ Z+ ff,
gg , fg, gf
4.1 f(n) = 2n+1
g(n) = 3n-1
4.2 f(n) = n2
g(n) = 2n

34. 1. 44 4
2. 35 7
( ABCD F)
3. 35,000
4
999
135
4. 6