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OnAAQNoNCGQBDLarSLQC

Study note

Chelsea

United States of America

DEFAULT (US)

DEFAULT

Arithmetic

Defining Functional Relationships 

Functions

Recognizing functions

One or More 3
Xs to One Y
Defining Functional Relationships
WARM UP
Evaluate each expression
given the set of values
{1, 6, 12, 25).
1. 5x
2. 2x+
+1
3. x8
LEARNING GOALS
• Describe a functional relationship in terms of a rule which
assigns to each input exactly one output.
• Determine whether a relation (represented as a mapping,
set of ordered pairs, table, sequence, graph, equation, or
context) is a function.
KEY TERMS
mapping
• set
• relation
input
• output
• function
domain
●
range
• scatter plot
vertical line test
Throughout middle school, you have investigated different types of relationships between
variable quantities: additive, multiplicative, proportional, and non-proportional. What are
functional relationships?
LESSON 3: One or More Xs to One Y M2-205 Getting Started
1. Write an equation to describe the relationship between each
X→ [rule] →y "independent variable x and the dependent variable y. Explain
your reasoning.
M2-206
What's My Rule?
Rules can be used to generate sequences of numbers. They can also
be used to generate (x, y) ordered pairs.
y=4x+12
= 4(3) +12
=12+12
= 24 ✓
se
You can
sketch the
graph to help
determine the
equation.
99
a.
C.
X
y
-12
0
12
24
12-0 12
0-(-3) 3
m =
-6
-3
0
3
X
-10
-2
0
5
S
● TOPIC 3: Introduction to Functions.
y
9
1
-1
4
=4
b.
d.
X
1
5
-1
-10
X
0
4
5
20
y
-2
-10
2
20
y
2
4
4.5
12
m=4-2=2=15 ㅗ
2. Create your own table and have a partner determine the
equation you used to build it.
२
y=/x+2 ACTIVITY
3.1
As you learned previously, ordered pairs consist of an x-coordinate
and a y-coordinate. You also learned that a series of ordered pairs
on a coordinate plane can represent a pattern. You can also use
a mapping to show ordered pairs. A mapping represents two
sets of objects or items. Arrows connect the items to represent a
relationship between them.
When you write the ordered pairs for a mapping, you are
writing a set of ordered pairs. A set is a collection of numbers,
geometric figures, letters, or other objects that have some
characteristic in common.
Functions as Mappings from
One Set to Another
1. Write the set of ordered pairs that represent a relationship in
each mapping.
a.
*
5
{(1,7) (0, 1) (3,5) (4,3)}
C.
_ ო ი |
1895
7
b.
5
7
8
{(1, 1) (2,3) (2,5) (3,5)(4,9) (5,9)}
5800
d.
2. Create a mapping from the set of ordered pairs.
a. (5.8). (11.7.685)
+9
4
5.
3
+9
5
7
9
4
17
78
20
b. {(3,4) (9, 3). (67), (4, 20)
x
Use braces, {}, to
denote a set.
{(1, 1) (2, 1) (3,5) (4,3) (57)
{(2,7) (a,20) (4,9) (59)(82)
LESSON 3: One or More Xs to One Y M2-207 (x, y)
↓ ↓
imput output
ce
3. Write the set of ordered pairs to represent each table.
domain
range
(x-values) (y-values)
a.
Notice the
use of set
notation when
writing the
domain and
range.
Input
-10
-5
0
5
Output
-20
-10
0
10
20
b.
2
M2-208 ● TOPIC 3: Introduction to Functions
X
4
20
10
{(-10,~20) (~5,-10)(0,0) (5,10) (10,00)}
10
0
10
20
In each mapping shown, the domain is {1, 2, 3, 4}.
Seta group
of ordered pairs.
The mappings and ordered pairs shown in Questions 1 through 3
form relations. A relation is any set of ordered pairs or the mapping
between a set of inputs and a set of outputs. The first coordinate of
an ordered pair in a relation is the input, and the second coordinate is
the output. A function maps each input to one and only one output.
In other words, a function has no input with more than one output.
The domain of a function is the set of all inputs of the function. The
range of a function is the set of all outputs of the function.
*
One x to one y
WORKED EXAMPLE
y
-10
-5
7
0_5_100
2
O
The range is {1, 3, 7}.
The range is {1, 3, 5, 7}.
Each mapping represents a function because no input, or domain
value, is mapped to more than one output, or range value. WORKED EXAMPLE
In the mapping shown, the domain is {1, 2, 3, 4, 5} and the range is
{1, 3, 5, 7).
5
3
5
This mapping does not represent a function.
7
4. State why the relation in the worked example shown is not
a function.
Not a function because
not one-to-one
#2@ Domain: {5,6,8,11}
Range: {5,8,9}
5) Domain: {3,4,9}
(2,3
5
(2,5)
5. State the domain and range for each relation in Questions 2
and 3. Then, determine which relations represent functions.
If the relation is not a function, explain why not.
Function.
Not a function
5 Domain: {0, 10,20}
3
Range: {4,7,8,20}
#3 Domain: {-10, ~5,0, 5, 10}
Range: {-20, -10, 0, 10,20}
Range: -10,-5,0,5,10}
Function
5
7
No
5:
Not a fonction
NOTES
yes
one-forone
18 not
9 one-to-one
>10
LESSON 3: One or More Xs to One Y M2-209 Ge
Think about
the mappings
as ordered
pairs.
se
Remember
that a
sequence has
a term number
and a term
value.
99
6. Review and analyze Emil's work. Explain why Emil's mapping is
not an example of a function.
Emil
My mapping represents a function.
b. 1, 0, 1, 0, 1, ...
c. 0, 5, 10, 15, 20, ...
9
M2-210 • TOPIC 3: Introduction to Functions
12
Not a function
because
4
→ 3
5
7. Determine if each sequence represents a function. Explain why
or why not. If it is a function, identify its domain and range.
Create a mapping to verify your answer.
a. 2, 4, 6, 8, 10, ...
maps
to 3&5
4= Yes
No
Yes
No
ACTIVITY
3.2
No
You have determined if sets of ordered pairs represent functions.
In this activity you will examine different situations and determine
whether they represent functional relationships.
Read each context and decide whether it fits the definition of a
function. Explain your reasoning.
Functions as Mapping
Inputs to Outputs
3. Input: There are four puppies in a litter.
Yes Output: One puppy was adopted by the Smiths, another by
the Jacksons, and the remaining two by the Fullers.
1. Input: Sue writes a thank-you note to her best friend.
Output: Her best friend receives the thank-you note in the mail.
2. Input: A football game is being telecast.
Output: It appears on televisions in millions of homes.
4. Input:
Output:
5. Input:
Output:
6. Input:
Pmc
The basketball team has numbered uniforms.
Each player wears a uniform with her assigned
number.
Beverly Hills, California, has the zip code 90210.
There are 34,675 people living in Beverly Hills.
A sneak preview of a new movie is being shown in a
local theater.
Output: 65 people are in the audience.
-834
2
homes
7149
DOOD
LESSON 3: One or More Xs to One Y M2-211 7. Input:
Output: Tara's job on any one day.
Yes
8. Input:
No
Output:
ACTIVITY
3.3
Tara works at a fast food restaurant on weekdays and
a card store on weekends.
M
то
Th
1.
10-
11
12-
13-
Janelle sends a text message to everyone in her
contact list on her cell phone.
Sat
There are 41 friends and family on Janelle's
contact list.
Sur
Analyze the relations in each pair. Determine which relations are
functions and which are not functions. Explain how you know.
Mapping A
Mapping B
Determining Whether
a Relation Is a Function
M2-212 • TOPIC 3: Introduction to Functions.
1000
2000
3000
Function
Domain: {10, 11, 12, 13}
Range: { 1000, 2000, 3000}
10-
11
1000
2000
3000
€
13.
Not a function
Domarh: {10, 11, 12, 13}
Range: {1000, 2000, 3000} 2.
Table A
Input
-2
-1
0
1
2
4
Domain: {-2,-1,0, 1, 2} Range: {0, 1,9}
3. Sequence A
Function.
Output
4
1
0
1
7, 10, 13, 16, 19, ...
4. Set A
{(2, 3), (2, 4), (2, 5), (2, 6), (2, 7)}
Not a function.
Domain: {2}
Range: {3,4,5,6,7
5. Scenario A
Input:
The morning
announcements
are read over the school
intercom system during
homeroom period.
Output:
All students report to
homeroom at the start of
the school day to listen
to the announcements.
X
Not a fanction
1 to 35
2
1
0
Table B
y
-4
-1
0
1
4
Not a function
Domain:
Sequence B
10, 30, 10, 30, 10, ...
{0,1,2}
Range
{-4,-1,0,1,4}
Set B
{(2, 1), (3, 1), (4, 1), (5, 1), (6, 1)}
Function
Domain: 2,3,4,5,6} Range:{1}
Scenario B
Input:
Each student goes
through the
cafeteria line.
Output:
Each student selects a
lunch option from the
menu.
Function.
LESSON 3: One or More Xs to One Y M2-213 A relation can be
represented as a
graph.
ACTIVITY
3.4
A scatter plot is a graph of a collection of ordered pairs that allows
an exploration of the relationship between the points.
1. Determine if each scatter plot represents a function.
Explain your reasoning.
a.
Output
Functions as Graphs
y
6
5
●
M2-214. TOPIC 3: Introduction to Functions
●
●
WORKED EXAMPLE
Consider the scatter plot shown."
In this scatter plot, the relation
is not a function. The input
value 4 can be mapped to
two different outputs, 1 and 4.
Those two outputs are shown
as intersections to the vertical
line drawn at x = 4.
2
1
O
0
0 1 2 3
5
{(1,1) (2,3) (3, 14,4)}
Function
The vertical line test is a visual method used to determine whether
a relation represented as a graph is a function. To apply the vertical
line test, consider all of the vertical lines that could be drawn on the
graph of a relation. If any of the vertical lines intersect the graph of
the relation at more than one point, then the relation is not a function.
9
8
7
6
5
4
3
2
1
0
b. YA
6
0
Output
4
2
3
2
1
0
0
1
2 3 4
5 6 X
Input
{(1, 1) (2, 3) (4,4) (4,1)}
Not a function
3
*
*
Not a
mc
fonction
8
9 2. Use the definition of function to explain why the vertical line
test works.
Vertical line test - A relation is a
function if the vertical line crosses
each paint one at a time.
3. Use the vertical line test to determine if each graph represents
a function. Explain your reasoning.
a.
y A
6
5
4
3
2
1
0
0
V
5
6
Nof a function.
Functions
b.
H,B, L,E
K, C, G, A
y A
6
5
4
3
2
1
0
N
2
3 4
Function.
5
4. Use the 12 cards that you sorted in the previous lesson.
Sort the graphs into two groups: functions and non-functions.
Use the letter of each graph to record your findings.
Non-functions
6
I, F, I, D
X
NOTES
LESSON 3: One or More Xs to One Y M2-215 NOTES
M2-216
ACTIVITY
3.5
Functions as Equations
So far, you have determined whether a mapping, context, or a graph
represents a function. You can also determine whether an equation
is a function.
WORKED EXAMPLE
The given equation can be used to convert yards to feet. Let x
represent the number of yards, and let y represent the number
of feet.
y = 3x
To test whether this equation is
a function, first, substitute values
for x into the equation, and then
determine if any x-value can
be mapped to more than one
y-value. If each x-value has exactly
one y-value, then it is a function.
Otherwise, it is not a function.
TOPIC 3: Introduction to Functions
X
1
3
4
8
y = 3x
3
9
12
24
In this case, every x-value can be mapped to only one y-value.
Each x-value is multiplied by 3. Some examples of ordered pairs
are (2, 6), (10, 30), and (5, 15). Therefore, this equation is a function.
It is not possible to test every possible input value in order to
determine whether or not the equation represents a function. You can
graph any equation to see the pattern and use the vertical line test to
determine if it represents a function. 1. Determine whether each equation is a function. List three
ordered pairs that are solutions to each. Explain your
reasoning.
a. y = 5x + 3
y=mx+b
linear equation /function
Function
c. y = |x|
Absolute value equation
Function V
e. y = 4
horizontal line
Function
X
b. y =
I
2. Explain what is wrong with Taylor's reasoning.
4
9
25
Y
2
Quadratic
(parabola)
U Function
3
5
d. x² + y² = 1
Circle
Not a function
Taylor
The equation y² = x represents a function.
f. x = 2
vertical line
Not a function
If you do not
recognize the graph
of the equation, use a
graphing calculator to
see the pattern.
se
If two different
inputs go to
the same
output, it
can still be a
function.
-99
LESSON 3: One or More Xs to One Y M2-217 NOTES
H
TALK the TALK
Function Organizer
1. Complete the graphic organizer for the concept of function.
Write a definition for function in your own words. Then,
create a problem situation that can be represented using a
function. Finally, create a table of ordered pairs and sketch
a graph to represent the function.
Definition
One x to
one у
(1-to-1)
Graph
M2-218 TOPIC 3: Introduction to Functions
discrete
- One capital to
one state
-one president to
one country
- one flavor to one
Function
Problem
Situation
cone.
- one license plate #
to one cav.
X-83+
continuous 4
على من وماما من
{(1,5) (2,5)(3,5) (4,5)}
Table/
Ordered Pairs Assignment
Write
Write the term from the box that best completes each sentence.
scatter plot
mapping
1. A(n).
is any set of ordered pairs or the mapping
between a set of inputs and a set of outputs.
2. The first coordinate of an ordered pair in a relation is the
3. The second coordinate of an ordered pair is the.
4. A(n)
5. A(n)
6. The
output relation input vertical line test
domain range function
set
maps each input to one and only one output.
is a graph of a collection of ordered pairs.
is a visual method of determining whether
a relation represented as a graph is a function by visualizing
whether any vertical lines would intersect the graph of the
relation at more than one point.
7. A(n)
shows objects in two sets connected
together to represent a relationship between the two sets.
8. A(n)
is a collection of numbers, geometric
figures, letters, or other objects that have some characteristic
in common.
9. The
function.
10. The
the function.
of a function is the set of all inputs of the
of a function is the set of all outputs of
Remember
A relation is any set of ordered
pairs or the mapping between
a set of inputs and a set of
outputs.
A relation is a function when
each input value maps to one
and only one output value.
Practice
1. A history teacher asks six of her students the number of hours that they studied for a recent test. The
diagram shown maps the grades that they received on the test to the number of hours that they studied.
a. Is the relation a function? If the relation is not a function, explain
why not.
b. Write the set of ordered pairs to represent the mapping.
c. What does the first value in each ordered pair in part (b) represent?
What does the second value in each ordered pair represent?
d. Create a scatter plot. Does the graph agree with your conclusion from
part (a)? Explain your reasoning.
85
70
95
Grade
6
Hours Studied
LESSON 3: One or More Xs to One Y M2-219 2. The science teacher created the set of ordered pairs {(100, 6), (90, 5), (80, 3), (70, 1), (90, 4), (80, 2)} to
represent six students' grades on the midterm to the number of hours that they had studied. Create a
mapping from this set of ordered pairs.
a. Is the relation a function? If the relation is not a function, explain why not.
b. List all the inputs of the relation.
c. List all the outputs of the relation.
d. Instead of mapping grades to hours studied, the teacher decides to create a new diagram.
This diagram maps hours studied to grades. Show the mapping that would result.
e. Write the set of ordered pairs to represent the mapping in part (d).
f. Is the relation in part (d) a function? If the relation is not a function, explain why not.
g. Create a scatter plot. Does the graph agree with your conclusion from part (f)? Explain your reasoning.
3. At the end of the year, a principal decides to create the given mapping.
Input: the 82 total students in the history class
Ouput: the final grades they received for the class
Does this mapping fit the definition of a function? Explain your reasoning.
4. Use the vertical line test to determine if each graph represents a function. Explain your reasoning.
b.
a.
-8
-6
-2
8-
19
4
2
-4-
-6-
-8-
y
2
4
6 8
8
M2-220 ● TOPIC 3: Introduction to Functions
6
4
2
AY
-4 -20
-2
-4
-6-
-8-
Stretch
Describe how you can tell from an equation whether a function is increasing, decreasing, or constant.

Defining Functional Relationships

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Defining Functional Relationships

Similar Content

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying