1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5. 2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given. 3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.

1. T.KAINOY
(function)
x1,y1) ∈ r x1,y2) ∈
r y1= y2
1.
1
r1 = {(0, 1),(1, 2),(1, 3),(2, 4)}
1 2 3
r2 = {(0, 1),(1, 2),(3, 1),(2, 4)}

2. T.KAINOY
(
)
2.
r = {(x,y) ∈ A× B | P(x,y) } x
P(x,y) y x y 1
r3 = {(x, y) | y2 = x } x=4
y=2 -2
r4 = {(x, y) | y = x2 } x
y
r5= {(x, y) | y = x } x=2
y = 2, -2
3.
y
1

3. T.KAINOY
r (x , y1) r
(x , y2) r y1 = y2
2 r = {(x,y) R R y2 = 4x +1
}
y2 = 4x + 1
(x , y1) r y12 = 4a + 1 ….. (1)
(x , y2) r y22= 4a + 1 ….. (2)
(1) (2) y12 = y22
y1 = y2
y1 = y2
3 r = {(x,y) R R y= x 1}
y= x 1
(x , y1) r y1 = a 1 …..(1)
(x , y2) r y2 = a 1 …..(2)
(1) (2) y1 = y2

4. T.KAINOY
y = f(x) y = g(x)
f ( x, y) R Ry 2x 5 y 2x 5
f ( x) 2x 5
g ( x, y) R Ry x2 y x2
g ( x) x2
f(x) f x
4 f f ( x) 2 x 2 1
f(0) , f(2) f(-1)
f ( x) 2 x 2 1
f (0) 2(0) 2 1 1
f (2) 2(2) 2 1 7
f ( 1) 2( 1) 2 1 1
5 f (1) 2 f ( x 1) 1
2
x
f ( x)
f (4)
2
f ( x 1) 1 f (1) 2
f ( x)
2 2
x 1 f (2) 1 1 2
f (1) 2
2 2
x 2 f (3) 1 1 2
f (2) 2
2 2
x 3 f (4) 1 1 2
f (3) 2
1. A B (f : A B)

5. T.KAINOY
Df = A Rf B
6 A = {1,2,3,4} B = {3,6,7,8}
1. f1 = { (1,3) , (2,6) , (3,7) , (4,8) }
f1 D f1 = { 1,2,3,4, } = A
f1 A B
2. f2 = { (1,6) , (2,7) , (3,8) }
f2 D f2 = { 1,2,3 } A
f2
f2 D f2 B
2. A B (f : A onto
B)
Df = A Rf = B
3. 1 – 1 (One – to – one function )
f 1-1
2

6. T.KAINOY
f
A B A B
f1
1 x f2
m
x
2 y
o
y
3 z
n
f1 1-1
f2 1-1
many-to-one
f 1–1
1.
X
X
1
1–1
2. X

7. T.KAINOY
1-1
many – to – one
3.
(x1,y) f (x2,y) f
x1 = x 2
1-1
7 f f ={(x , y) R R
X 1 + Y 1 =2}
f 1-1
(x1,y) f (x2,y) f
x1 1 + y 1 =2 ….. (1)
x2 1 + y 1 =2 …...(2)
(1)=(2) x1 1 = x2 1
x1 +1= x2 +1
x1 = x2
1-1
8 f = {(x , y) R R y = x 2}
f 1-1
(x1,y) f (x2,y) f
y = x 12 …….. (1)
c = x 22 …….. (2)
x12= x22 x1 = x2

8. T.KAINOY
x1 = x2
f 1-1
y
x y
4. A B” (f:A 1 1
B)
Df = A Rf B “ y
x ”
5. A B (f:A B)
Df = A Rf = B “ y
x ”
f A B f
4

9. T.KAINOY
1. (Constant Function)
f (x) = a ( )
f (x) = 2 , f (x) = -3
2. (Linear Function)
f (x) = ax + b ( )

10. T.KAINOY
f (x) = 5x+3 , f (x) = 4x
3. (Quadratic Function)
f (x) = ax2+ bx + c ( )
f (x) = 3x2+ 2x + 1 , f (x) = 7x2- 4
4. (Polynomial Function)
f(x) = an x n an 1 x n 1
... a2 x 2 a1 x a0
an , an 1 ,..., a2 , a1 , a0 n
f(x) = 2x5+ 3x3 + 4x + 7
5. (Rational Function)
f(x) = p(x), q(x) q(x) 0
f(x) = 3x 2
x2 1
6. (Absolute Value Function)
f (x) = ax + b + c ( )
f(x) x
7. step function)

11. T.KAINOY
8. periodic function)
f p f(x+p)
= f(x) x x+p f