1. 2.2 An Introduction to Functions (الدوال)

2. Definitions:
1) Any set of ordered pairs (x, y) is called a relation(علاقة)
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.

3. Ex1: Determine whether the relation represents y as a function of x.
b) {(-1, 1), (-1, -1), (0, 3), (2, 4)}
Not a Function

4. 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}

5. Function notation
Functions represented by equations are often named using a letter such as f, g, h, ….
To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function.

6. Ex3:

7. Case1:Algebraically Case 2:Graphically
How to identify if an equation represents a function or not?
جبرياَ
Case1:Algebraically
Solve for y
two values of y
not function
one value of y
function
Case 2:Graphically
من الرسم
The 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.

8. Ex4:
Identify which one of the following relations define y as a function of x

9. Ex5:
Identify which of the following graphs are graphs of functions
y 2 = x x y
y 2 = x x y
y = x 2 x y
y = x 2 x y
x 2 + y 2 = 1 x y b) c) a)
function
Not function
Not function

10. d) e) f)
x = | y – 2|
y=1
x=1 x
function
Not function
Not function

11. 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)}

12. b) y = x3 + 3x2 – x – 1 a) y = x3 one-to-one not one-to-one
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.
b) y = x3 + 3x2 – x – 1
a) y = x3 x y -4 4 8 x y -4 4 8
one-to-one
not one-to-one

13. 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?
Definition of Domain

14. Ex8: Find the domain of the following functions

16. Increasing, Decreasing, and Constant
ثابت تناقص تزايد x y b c a d
● Increases on [c, d]
● Decreases on [a, b]
●Constant on [b, c]
Rules:
Start from left to right
Take the intervals from the x-axis
Example: Find Domain

17. Consider the graph of the function f(x)
(3, -4) x y
(-3, 6)
Ex9:
Consider the graph of the function f(x)
This graph is
● Increasing on (-∞, -3]
[3, +∞).
● Decreasing on [-3, 3]

18. Piecewise-defined function
EX10:

19. The Greatest Integer Function ( Floor Function)
Ex11: Find the value of

21. Find the value of f(-5)+f(5) Let
Ex12
Find the value of f(-5)+f(5)
Let

22. Solve the following equations
Notes:
Ex13
Solve the following equations

23. Ex14:HW
Find the domain of the function
Ex15:HW
Find the x-intercept and y-intercept of

24. Ex16:
Sketch the graph of
Q43/191
Sketch the graph of
The End