Module 3.2 Understanding Relations And...

Module 3.2

Understanding RelationsAnd Functions

How do you represent relations and functions?

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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.

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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.

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?

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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.

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Domain Range

The given relation represents the amount of gas in gallons and the distancetraveled in miles from that amount of gas.

Table Mapping Diagram Graph

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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.

The relation represents the number of books sold and the price for thecorresponding number of books.


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The relation represents the time spent exercising and the number of caloriesburned during that time.


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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) }


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.



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.


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.

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.

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.

Module 3.2 Understanding Relations And...

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