1. Chapter 2: Functions and Graphs
PART 1

2. FlasBack… What the correlation ?
Define the relationships that are presented.
Husband and Wife
Father and son
Foot length shoe size
Number of drinks and BAC

3. There are other ways to describe relations between variables.
 Set to set
Ordered pairs

4. A set of ordered pairs (x, y) is also called a relation.
The domain is the set of x-coordinates of the ordered pairs.
The range is the set of y-coordinates of the ordered pairs.

5. Example 1
Find the domain and range of the relation
{(4,9), (-4,9), (2,3), (10,-5)}
Domain is the set of all x-values, {4, -4, 2, 10}.
Range is the set of all y-values, {9, 3, -5}.

6. Example 2 Domain: Range: {20, 15, 10, 7}
{Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}
Range:
{20, 15, 10, 7}

7. Some relations are also functions.
A function is a set of order pairs in which each first component in the ordered pairs corresponds to exactly one second component.

8. f FUNCTION CONCEPT x DOMAIN y RANGE
A function f is a set of ordered pairs (x,y) where each x-value corresponds to exactly one y-value.
FUNCTION CONCEPT

9. R NOT A FUNCTION y1 y2 RANGE x DOMAIN
A function f is a set of ordered pairs (x,y) where each x-value corresponds to exactly one y-value.
DOMAIN
NOT A FUNCTION

10. f FUNCTION CONCEPT x1 DOMAIN x2 y RANGE
A function f is a set of ordered pairs (x,y) where each x-value corresponds to exactly one y-value.
FUNCTION CONCEPT

11. Ways to Represent a Function
Symbolic
Graphical
Numeric
X Y
5 10
-1 -2
Verbal The cost is twice the original amount.

12. FUNCTION NOTATION
• Input Value • Member of the Domain • Independent Variable
These are all equivalent names for the x.
• Output Value • Member of the Range • Dependent Variable
These are all equivalent names for the y.
Name of the function
One of the really big deals is to remember that y is f(x). That means that f(x) and y are interchangeable.

13. Example 3
y = -3x + 2 so  represents a function.
We often use letters such as f, g, and h to name functions.
We can use the function notation f(x) (read “f of x”) and write the equation as f(x) = -3x + 2.
Note: The symbol f(x) is a specialized notation that does NOT mean f • x (f times x).

14. to evaluate a function at x substitute the x-value into the notation.
Example 4
f(x) = -3x + 2, f(2) = -3(2) + 2 = = -4.

15. write down the corresponding ordered pair.
Example 5
g(x) = x2 – 2x
find g(-3)
write down the corresponding ordered pair.
Answer :
g(-3) = (-3)2 – 2(-3) = 9 – (-6) = 15.
The ordered pair is (-3, 15).

16. Drawing Graphs of Functions
A way to visualize a function is by drawing its graph
The graph of a real function f of one variable is the set of all points P(x, y) in the plane such that y = f(x).
Plot the value of x on the horizontal, or x-axis and the value of f(x) on the vertical, or y-axis.
How can we tell whether a set of points in the plane is the graph of some function? By reading the definition of a function again, we have an answer.

17. Graphs of functions??

19. Domain and Range
Ex 6.
Given the relation {(4,9), (-4,9), (2,3), (10,-5)}, is it a function?
Since each element of the domain (x-values) is paired with only one element of the range (y-values) , it is a function.
Note:
Each x-value has to be assigned to ONLY
one y-value!!!

20. Is the relation y = x2 – 2x a function?
Example 7
Is the relation y = x2 – 2x a function?
Since each element of the domain (the x-values) would produce only one element of the range (the y-values), it is a function.
Question:
What does the graph of this function look like?
Does this graph pass
the vertical line test?

21. Is the relation x2 + y2 = 9 a function?
Example 8
Is the relation x2 + y2 = 9 a function?
Since each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function
Check the ordered pairs: (0, 3) (0, -3)
The x-value 0 corresponds to two different y-values, so the relation is NOT a function.
Question: What does the graph of this relation look like?

22. Ex
Range
How to see the domain
Domain

23. Example 9 x y
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since no vertical line will intersect this graph more than once, it is the graph of a function.

24. Example 10 x y
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since no vertical line will intersect this graph more than once, it is the graph of a function.

25. Example 11 x y
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since a vertical line can be drawn that intersects the graph at every point, it is NOT the graph of a function.

26. Example 12 x y
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.

27. Domain is [-3, 4] Range is [-4, 2]
Determining the domain and range from the graph of a relation:
Example: x y
Find the domain and range of the function graphed (in red) to the right. Use interval notation.
Domain is [-3, 4]
Domain
Range is [-4, 2]
Range

28. Example 13 x y
Find the domain and range of the function graphed to the right. Use interval notation.
Range is [-2, )
Range
Domain is (-, )
Domain

29. Example 14 x y
Find the domain and range of the function graphed to the right. Use interval notation.
Domain: (-, )
Range: (-, )

30. (The range in this case consists of one single y-value.)
Example 15 x y
Find the domain and range of the function graphed to the right. Use interval notation.
Domain: (-, )
Range: [-2.5]
(The range in this case consists of one single y-value.)

31. Domain: [-4, 4] Range: [-4.3, 0] Example 16 y
Find the domain and range of the relation graphed to the right. Use interval notation.
(Note this relation is NOT a function, but it still has a domain and range.)
Domain: [-4, 4]
Range: [-4.3, 0]

32. Domain: [2] Range: (-, ) Example 17 y
Find the domain and range of the relation graphed to the right. Use interval notation.
(Note this relation is NOT a function, but it still has a domain and range.)
Domain: [2]
Range: (-, )

33. Decide the Domain and Range: Graph Homework1

34. 2

35. 3

36. Decide the Domain and Range: Graph