1) Any set of ordered pairs (x, y) is called a relation(عالقة)

Definitions:

)}1,2(),0,2(,)1,1{(:ex

2) A function (دالة) is a relation such that for every first coordinate x in the ordered pair there is only one 2nd coordinate y. i.e. No repetition in the first coordinate.

Ex1: Determine whether the relation represents y as a function of x.

a) {(-2, 3), (0, 0), (2, 3), (4, -1)}

b) {(-1, 1), (-1, -1), (0, 3), (2, 4)}

Function

Not a Function

3)The domain (المجال) of the function is the set of all first coordinate of the ordered pairs.

4) The range (المدى) of the function the set of all second coordinate of the ordered pair.

Ex2: Consider the function {(0,0) , (-1,1), (2,3) }

The domain is {0,-1,2}

The range is {0,1,3}

• Functions represented by equations are often named using a letter such as f, g, h, ….

• Function notation

• To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function.

Ex3:

hxfhxf

d

afc

fb

fa

xxf

)()(.

)2(.

)0(.

)1(.

Find.22)(Let 2

How to identify if an equation represents a function or not?

Solve for y

two values of y not function

one value of y function

Case1:Algebraically

Case 2:GraphicallyThe vertical (العمودي) line test : A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

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Ex4:

Identify which one of the following relations define y as a function of x

xyd

xyc

yxb

yxa

)

)

23)

4)

23

22

Ex5:

Identify which of the following graphs are graphs of functions

y 2 = x

x

y

x 2 + y

2 = 1

x

y

y = x 2

x

y

Not function function

y 2 = x

x

y

y = x 2

x

ya)

b) c)

Not function

d) e)

x

x = | y – 2|

Not function functionNot function

f)

x=1 y=1

Def.:

A one-to-one function is a function such that for every y, there is only one x that can be paired with y .[i.e. No repetition in either x or y]

Ex6: Determine whether the relation represents y as

a function of x.

a){(1,2), (2,2), (3,4)}

b){(1,1), (1,3), (4,5)}

c){(1,1), (2,2), (3,0)}

• Horizontal (أفقي) Line Test

A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point.

Ex7: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one.

a) y = x3

x

y

-4 4

4

8

one-to-one not one-to-one

x

y

-4 4

4

8

b) y = x3 + 3x2 – x – 1

• The domain(المجال) of a function f is the set of all real numbers for which the function makes sense.

How to find the domain?

0root square under the2)

0rdenominato the)1

Ex8: Find the domain of the following functions

units is radius whose

sphere a of volume theis )( where,31

)()4

2)()3

21

)()2

53)1

3

r

rVrrV

xxf

xxf

x f (x)

)3)(2(1

)()9

1)()8

1)((HW))7

)((HW))6

4)()5

3

2

2

xxx

xf

xxw

xxf

xxxh

xxg

• Increasing, Decreasing, and Constant

x

y

b ca d

Rules:

1) Start from left to right

2) Take the intervals from the x-axis

● Increases on [c, d]

● Decreases on [a, b]

●Constant on [b, c]

تناقص • ثابت تزايد

This graph is

● Decreasing on [-3, 3]

● Increasing on (-∞, -3] [3, +∞).

(3, -4)

x

y(-3, 6)

Ex9: Consider the graph of the function f(x)

•Piecewise-defined function

4 such that of valuesallfind)6

)( of range theanddomain the find 5)

constant and ,decreasing ,increasing of (s) interval the find 4)

intercepttheandinterceptthedetermine)3

)6(),3(),0(),2(find)2

)(ofgraph thesketch)1

511

12

23

)(Let 2

f(x)x

xf

yx

ffff

xf

xifx

xifx

xif

xf

EX10:

The Greatest Integer Function ( Floor Function)

xxxx integergreatest the)int(

5.1)4

3.1)3

7.1)2

5.1)1

Ex11: Find the value of

)10

0)9

2)8

2)7

3.1)6

7.1)5

Ex12

034

031

)(

xifx

xifxxf

Find the value of f(-5)+f(5)

Let

Notes: integer:,1)()()1 nnxfnnxf

integersallofsetthe

),(then ,)(If)2

f

f

R

Dxxf

Ex13 Solve the following equations

212)1 x

21

12)2 x

11

)(

x

xf

Ex14:HW

Find the domain of the function

Ex15:HW

Find the x-intercept and y-intercept of

312

)(x

xf

Ex16:

Sketch the graph of xxf )(

Q43/191

Sketch the graph of 66for31

)(

xxxf