1-6 relations and functions1 relations and functions section 1-6
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1-6 Relations and Functions 1
Relations and Functions
Section 1-6
04/21/23 12:30 1-6 Relations and Functions 2
Review A relation between two variables x and
y is a set of ordered pairs An ordered pair consist of a x and y-
coordinate A relation may be viewed as ordered pairs,
mapping design, table, equation, or written in sentences
x-values are inputs, domain, independent variable
y-values are outputs, range, dependent variable
04/21/23 12:30 1-6 Relations and Functions 3
Example 1
{( , ), ( , ), ( , ), ( , ), ( , ), ( , )} 0 5 1 4 2 3 3 2 4 1 5 0
•What is the domain? {0, 1, 2, 3, 4, 5}What is the range?
{-5, -4, -3, -2, -1, 0}
04/21/23 12:30 1-6 Relations and Functions 4
Example 2
•What is the domain? {4, -5, 0, 9, -1}
•What is the range?{-2, 7}
Input 4 –5 0 9 –1
–2 7Output
04/21/23 12:30 1-6 Relations and Functions 5
Is a relation a function?
What is a function?
According to the textbook, “a function is…a relation in
which every input is paired with exactly one output”
04/21/23 12:30 1-6 Relations and Functions 6
Is a relation a function?•Focus on the x-coordinates, when given a relation
If the set of ordered pairs have different x-coordinates, it IS A function
If the set of ordered pairs have same x-coordinates, it is NOT a function
•Y-coordinates have no bearing in determining functions
04/21/23 12:30 1-6 Relations and Functions 7
Example 3
YES
{( , ), ( , ), ( , ), ( , ), ( , ), ( , )} 0 5 1 4 2 3 3 2 4 1 5 0
•Is this a function?•Hint: Look only at the x-coordinates
04/21/23 12:30 1-6 Relations and Functions 8
Example 4
{(– , ), ( , ), ( , ), ( , ), ( , ), (– , )} 1 7 1 0 2 3 0 8 0 5 2 1
•Is this a function?•Hint: Look only at the x-coordinates
NO
04/21/23 12:30 1-6 Relations and Functions 9
Choice One Choice Two
Example 5
310
–123
2–13
23–20
Which mapping represents a function?
Choice 1
04/21/23 12:30 1-6 Relations and Functions 10
Example 6Which mapping represents a function?
A. B.
B
04/21/23 12:30 1-6 Relations and Functions 11
Example 7Which situation represents a function?
There is only one price for each different item on a certain date. The relation from items to price makes it a function.
A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function.
a. The items in a store to their prices on a certain date
b. Types of fruits to their colors
04/21/23 12:30 1-6 Relations and Functions 12
Vertical Line Test•Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point.
AKA: “The Pencil Test”Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function
04/21/23 12:30 1-6 Relations and Functions 13
Vertical Line Test
Would this graph be a function?YES
04/21/23 12:30 1-6 Relations and Functions 14
Vertical Line Test
Would this graph be a function?
NO
04/21/23 12:30 1-6 Relations and Functions 15
Is the following function discrete or continuous? What is the Domain? What is the Range?
Discrete-7, 1, 5, 7, 8, 10
1, 0, -7, 5, 2, 8
04/21/23 12:30 1-6 Relations and Functions 16
Is the following function discrete or continuous? What is the Domain? What is the Range?
continuous8,8
6,6
04/21/23 12:30 1-6 Relations and Functions 17
Is the following function discrete or continuous? What is the Domain? What is the Range?
continuous0,45
10,70
04/21/23 12:30 1-6 Relations and Functions 18
Is the following function discrete or continuous? What is the Domain? What is the Range?
discrete-7, -5, -3, -1, 1, 3, 5, 7
2, 3, 4, 5, 7
04/21/23 12:30 1-6 Relations and Functions 19
Domain and Range in Real Life
The number of shoes in x pairs of shoes can be expressed by the equation y = 2x.
What subset of the real numbers makes sense for the domain?
Whole numbers
What would make sense for the range of the function?
Zero and the even numbers
1-6 Relations and Functions 20
Domain and Range in Real Life
The number of shoes in x pairs of shoes can be expressed by the equation y = 2x.
What is the independent variable?
The # of pairs of shoes.
What is the dependent variable?
The total # of shoes.
1-6 Relations and Functions 21
Domain and Range in Real LifeMr. Landry is driving to his hometown. It takes four
hours to get there. The distance he travels at any time, t, is represented by the function d = 55t (his average speed is 55mph.
Write an inequality that represents the domain in real life.
Write an inequality that represents the range in real life.
0 4x
0 220y
04/21/23 12:30 1-6 Relations and Functions 22
Domain and Range in Real LifeMr. Landry is driving to his hometown. It takes four
hours to get there. The distance he travels at any time, t, is represented by the function d = 55t (his average speed is 55mph.
What is the independent variable?
What is the dependent variable?
The time that he drives.
The total distance traveled.
04/21/23 12:30 1-6 Relations and Functions 23
Domain and Range in Real LifeJohnny bought at most 10 tickets to a concert for him
and his friends. The cost of each ticket was $12.50. Complete the table below to list the possible domain and range.
1 2 3
12.50
25.00
37.50
4
50
5
62.50
6 7 8 9 10
75 125
112.50
10087.50
The number of tickets bought.
What is the dependent variable?
The total cost of the tickets.
What is the independent variable?
04/21/23 12:30 1-6 Relations and Functions 24
Domain and Range in Real LifePete’s Pizza Parlor charges $5 for a large pizza with no
toppings. They charge an additional $1.50 for each of their 5 specialty toppings (tax is included in the price).
Jorge went to pick up his order. They said his total bill was $9.50. Could this be correct? Why or why not?
Yes
Susan went to pick up her order. They said she owed $10.25. Could this be correct? Why or why not?
No
One pizza with 3 toppings cost $9.50
One pizza with 4 toppings cost $11
04/21/23 12:30 1-6 Relations and Functions 25
Domain and Range in Real LifePete’s Pizza Parlor charges $5 for a large pizza with no
toppings. They charge an additional $1.50 for each of their 5 specialty toppings (tax is included in the price).
What is the independent variable?
The number of toppings
What is the dependent variable?
The cost of the pizza
04/21/23 12:30 1-6 Relations and Functions 26
Function Notationf(x) means function of x and is read “f of x.”
f(x) = 2x + 1 is written in function notation. The notation f(1) means to replace x with 1 resulting in the function value.
f(1) = 2x + 1 f(1) = 2(1) + 1 f(1) = 3
04/21/23 12:30 1-6 Relations and Functions 27
Function NotationGiven g(x) = x2 – 3, find g(-2) .
g(-2) = x2 – 3g(-2) = (-2)2 – 3g(-2) = 1
04/21/23 12:30 1-6 Relations and Functions 28
Function NotationGiven f(x) = , the following.
f(3) = 2x2 – 3xf(3) = 2(3)2 – 3(3)f(3) = 2(9) - 9f(3) = 9
a. f(3) b. 3f(x) c. f(3x)
3f(x) = 3(2x2 – 3x)3f(x) = 6x2 – 9x
f(3x) = 2x2 – 3xf(3x) = 2(3x)2 – 3(3x)f(3x) = 2(9x2) – 3(3x)f(3x) = 18x2 – 9x
22 3x x
04/21/23 12:30 1-6 Relations and Functions 29
For each function, evaluate f(0), f(1.5), f(-4),
f(0) =
f(1.5) =
f(-4) =
3
4
4
04/21/23 12:30 1-6 Relations and Functions 30
For each function, evaluate f(0), f(1.5), f(-4),
f(0) =
f(1.5) =
f(-4) =
1
3
1
04/21/23 12:30 1-6 Relations and Functions 31
For each function, evaluate f(0), f(1.5), f(-4),
f(0) =
f(1.5) =
f(-4) =
-5
1
1
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