The document defines and provides examples of different types of functions: 1. Constant functions where f(x) = a for all values of x (e.g. f(x) = 2). 2. Linear functions of the form f(x) = ax + b (e.g. f(x) = 5x+3). 3. Quadratic functions of the form f(x) = ax2 + bx + c (e.g. f(x) = 3x2 + 2x + 1).

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 R y 2x 5 y 2x 5
f ( x) 2x 5
2 2
g ( x, y ) R R y x y x
2
g ( x) x
f(x) f x
4 f f ( x) 2x
2
1
f(0) , f(2) f(-1)
2
f ( x) 2x 1
2
f (0) 2(0) 1 1
2
f (2) 2(2) 1 7
2
f ( 1) 2 ( 1) 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 1x
n 1
... a2 x
2
a1 x a0
a n , a n 1 ,..., a 2 , a 1 , a 0 n
f(x) = 2x5+ 3x3 + 4x + 7
5. (Rational Function)
f(x) = p(x), q(x) q(x) 0
f(x) = 3x
2
2
x 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