The document defines and provides examples of several types of functions including: 1) Constant functions where f(x) = a for all values of x. 2) Linear functions of the form f(x) = ax + b. 3) Quadratic functions of the form f(x) = ax2 + bx + c. 4) Polynomial functions which are the sum of terms with variables raised to various powers.

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
r (x , y1) r
(x , y2) r y1 = y2

3. T.KAINOY
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 = y 2
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
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 Df2 = { 1,2,3 } A
f2
f2 Df2 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 =
x2 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) = 3 x
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