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Module 3.2
Understanding RelationsAnd Functions
How do you represent relations and functions?
P. 115
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Definitions:
A relation is a set of ordered pairs (x, y) where x is the input
value and y is the output value.
The domain is all possible inputs (or x-values) of a
relation.The range is all possible outputs (or y-values) of a
relation.
For example, the following relation represents the number of
whole-wheat cracker boxes sold and the money earned:
{ (1, 4) , (2, 8) , (3, 12) , (4, 16) }
1 box sold earned $4.2 boxes sold earned $8.3 boxes sold earned
$12.4 boxes sold earned $16.
The domain is all possible inputs: {1, 2, 3, 4} – Number of
whole-wheat cracker boxes soldThe range is all possible outputs:
{2, 8, 12, 16} – Money earned
In truth, relations and functions can be represented not only as
ordered pairs, but also as:1) A table2) A graph3) A mapping
diagram.
P. 115
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P. 115
For the following relation, the input, X, is the ages of boys,
and the output, Y, is their corresponding height, in inches.
Ordered Pairs Table Graph Mapping Diagram
{ (7, 41) ,(8, 45) , (9, 49) ,
(10, 52) ,(10, 53) , (11, 55) ,(12, 59) }
(Can be horizontal or vertical)
What is the domain of the relation, and what does it represent?
______________
What is the range of the relation, and what does it represent?
_______________________
The number 10 appears twice in the x column of the table. How
many times is it written in the domain?The domain is a list of the
input values and a number should only be listed once no matter how
many timesit actually appears in the relation.
In a mapping diagram, where do the arrows start, and where do
they point?They start at the inputs, or x-values, and point to the
outputs, or y-values.
Domain Range
In a mapping diagram, the arrows start at the inputs, or
x-values, and point to the outputs, or y-values.
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A function is a type of relation where for each input value
there is only one unique output value.
Example: 𝒚 = 𝒙𝟐 When x = 3, y will always be equal to 9.
Question: The map for the previous example appears below. Is it
a function?
P. 116
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The given relation represents the number of students and the
number of classroomsthe school has to have for the corresponding
number of students.
Ordered Pairs Table Graph Mapping Diagram
{ (40, 2) ,(45, 3) , (50, 4) }
What is the domain of the relation, and what does it represent?
____________________________________________
What is the range of the relation, and what does it represent?
_____________________________________________
This relation is a function, because each domain value is paired
with exactly one range value.
P. 116
Domain Range
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The given relation represents the amount of gas in gallons and
the distancetraveled in miles from that amount of gas.
Table Mapping Diagram Graph
P. 117
What is the domain, and what does it represent?
_______________________
What is the range, and what does it represent?
_______________________
For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.
This relation _______ a function. Each domain value is paired
with ________________range value.
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The relation represents the number of books sold and the price
for thecorresponding number of books.
Table Mapping Diagram Graph
P. 118
What is the domain, and what does it represent?
_______________________
What is the range, and what does it represent?
_______________________
For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.
This relation _______ a function. Each domain value is paired
with ________________range value.
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The relation represents the time spent exercising and the number
of caloriesburned during that time.
Table Mapping Diagram Graph
P. 118
What is the domain, and what does it represent?
_______________________
What is the range, and what does it represent?
_______________________
For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.For an input of ____________, there is an output of
____________.
This relation _______ a function. Each domain value is paired
with ________________range value.
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How can you tell if a relation is a function?
For an ordered pair or table:
Check whether there are two or more ordered pairs with the same
x-value and different y-values. If there are not, the relation is a
function.
Here’s a relation that’s a function: Here’s a relation that’s
not a function:
{ (1, 5) , ( 2, 3) , ( 3, 2) , ( 4, 1) , ( 5, 0) } { (7, 41) ,
(8, 45) , (9, 49) , (10, 52) , (10, 53) , (11, 55) , (12, 59) }
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How can you tell if a relation is a function?
For a mapping diagram:
If there is only one arrow starting at each x-value, the
relation is a function. If more than one arrow starts at the same
x-value, it is not a function.
Here’s a relation that’s a function: Here’s a relation that’s
not a function:
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How can you tell if a relation is a function?
For a graph:
A relation is a function if and only if a vertical line does not
pass through more than one point on the graph. This is called the
Vertical Line Test.
Here’s a relation that’s a function: Here’s a relation that’s
not a function:
Draw a vertical line through each point of the graph. Does any
line touch more than one point? If yes, it is NOT a function.
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Why does the vertical line test work?It shows if there is more
than one y-value for any x-value.
Can a relation be a function if a horizontal line passes through
more than one pointon its graph? Yes; if a horizontal line passes
through more than one point, it means that the same y-value
corresponds to more than one x-value. Each x-value could still have
only one y-value, so the relation could be a function.
Which representation – mapping diagram, table, or graph – makes
it easiest to determineif a relation is a function? Why?A graph is
easiest, because you can apply the vertical line test to a
graph.
To determine if the relation 𝒚 = 𝒙𝟐 − 𝟖𝒙 + 𝟒 is a function -
what would be the best way to represent this relation in order to
determine if it is a function or not?The domain of this relation is
all real numbers, which means there are an infinite number of
values. You cannot represent this with a table, a mapping diagram,
or a set of ordered pairs. The best way to represent this relation
would be with a graph. This method allows you to use the vertical
line test to determine if this relation is a function.
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Examine the mapping diagram. The first set is the months of the
year, and the second set is the possible number of days per month.
Is the relation a function? Explain.
