Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

A relation is any set of ordered pairs, $(x, y)$ .

A function is a special type of relation that maps every input, $x$ , to exactly one output, $y$ .

Tell if each relation is a function and explain.

1.

$x$	$y$
-3	3
-2	2
-1	1
0	0
1	1
2	2
3	3

Compare your answer:

Yes, this is a function because every input goes to exactly one output.

2.

This figure shows two table that each have one column. The table on the left has the header “Name” and lists the names “Jenny”, “R and y”, “Dennis”, “Emily”, and “Raul”. The table on the right has the header “Email” and lists the email addresses RHern and ez@state. edu, JKim@gmail.com, Raul@gmail.com, ESmith@state. edu, DBrown@aol.com, jenny@aol.com, and R and y@gmail.com. There are arrows starting at names in the name table and pointing towards addresses in the email table. The first arrow goes from Jenny to JKim@gmail.com. The second arrow goes from Jenny to jenny@aol.com. The third arrow goes from R and y to R and y@gmail.com. The fourth arrow goes from Dennis to DBrown@aol.com. The fifth arrow goes from Emily to ESmith@state. edu. The sixth arrow goes from Raul to RHern and ez@state. edu. The seventh arrow goes from Raul to Raul@gmail.com.

Compare your answer:

This relation is not a function. Raul and Jenny have 2 emails, so twice there are inputs that go to two outputs.

3.

This figure shows two table that each have one column. The table on the left has the header “Network” and lists the television stations “NBC”, “HGTV”, and “HBO”. The table on the right has the header “Program” and lists the television shows “Ellen Degeneres Show”, “Law and Order”, “Tonight Show”, “Property Brothers”, “House Hunters”, “Love it or List it”, “Game of Thrones”, “True Detective”, and “Sesame Street”. There are arrows that start at a network in the first table and point toward a program in the second table. The first arrow goes from NBC to Ellen Degeneres Show. The second arrow goes from NBC to Law and Order. The third arrow goes from NBC to Tonight Show. The fourth arrow goes from HGTV to Property Brothers. The fifth arrow goes from HGTV to House Hunters. The sixth arrow goes from HGTV to Love it or List it. The seventh arrow goes from HBO to Game of Thrones. The eighth arrow goes from HBO to True Detective. The ninth arrow goes from HBO to Sesame Street.

Compare your answer:

This is not a function because each input goes to multiple outputs.

4. $(- 3, 27), (- 2, 8), (- 1, 1), (0, 0), (1, 1), (2, 8), (3, 27)$

Compare your answer:

Each input, $x$ -value, is matched with only one output, $y$ -value. So this relation is a function.

5. $(9, - 3), (4, - 2), (1, - 1), (0, 0), (1, 1), (4, 2), (9, 3)$

Compare your answer:

The input ( $x$ -value), 9, is matched with two outputs ( $y$ -values), both 3 and –3. So this relation is not a function.

Self Check

Consider the two relations below. Which statement is true about both relations?

{ $(−3,−6),(−2,−4),(−1,−2),(0,0),(1,2),(2,4),(3,6)$ }
{ $(8,−4),(4,−2),(2,−1),(0,0),(2,1),(4,2),(8,4)$ }

Neither relation is a function.
Relation b is a function, but relation a is not a function.
Relation a is a function, but relation b is not a function.
Both relations are functions.

Additional Resources

Is the Relation a Function?

A relation is a set of ordered pairs. The set of the first components of each ordered pair are the inputs and the set of the second components of each ordered pair are the outputs. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

$(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)$

The set of inputs is { $1, 2, 3, 4, 5$ }. The set of outputs is { $2, 4, 6, 8, 10$ }.

Each input value is the independent variable, and is often labeled with the lowercase letter $x$ . Each output value, is the dependent variable, and is often labeled lowercase letter $y$ .

A function $f$ is a relation that assigns a single output to each input. In other words, no $x$ -values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each input, {1,2,3,4,5}, is paired with exactly one output, { $2, 4, 6, 8, 10$ }.

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

{ (odd,1),(even,2),(odd,3),(even,4),(odd,5) }

Notice that each input, {even,odd} is not paired with exactly one output, { $1, 2, 3, 4, 5$ }. For example, the term “odd” corresponds to three output values, {1,3,5} and the term “even” corresponds to two output values, { $2, 4$ }. This violates the definition of a function, so this relation is not a function.

A mapping is sometimes used to show a relation. The arrows show the pairing of the inputs with the outputs.

Here are some relations that are functions and not functions.

Three diagrams show input-output mappings. (a) and (b) show functions: each input (p, q, r) points to one output. (c) is not a function, as input q points to two outputs (y and z).

This relationship is a function because each input is associated with a single output. Note that inputs $q$ and $r$ both give output $n$ .
This relationship is also a function. In this case, each input is associated with a single output.
This relationship is not a function because input $q$ is associated with two different outputs.

How to determine if a relationship is a function.

Identify the input values.
Identify the output values.
If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Try it

Try It: Is the Relation a Function?

Is the following relation a function? Explain your answer.

This figure shows two table that each have one column. The table on the left has the header “Name” and lists the names “Rebecca”, “Jennifer”, “John”, “Hector”, “Luis”, “Ebony”, “Raphael”, “Meredith”, “Karen”, and “Joseph”. The table on the right has the header “Birthday” and lists the dates “January 18”, “February 15”, “April 1”, “April 7”, “June 23”, “July 30”, “August 19”, and “November 6”. There are arrows starting at names in the Name table and pointing towards dates in the Birthday table. The first arrow goes from Rebecca to January 18. The second arrow goes from Jennifer to April 1. The third arrow goes from John to January 18. The fourth arrow goes from Hector to June 23. The fifth arrow goes from Luis to February 15. The sixth arrow goes from Ebony to April 7. The seventh arrow goes from Raphael to November 6. The eighth arrow goes from Meredith to August 19. The ninth arrow goes from Karen to August 19. The tenth arrow goes from Joseph to July 30.

Here is how to determine if this relation is a function:

Since every input (name) goes to only one output (birthday), this relation is a function.

4.1.3 Examining Relations and Functions

Activity

Additional Resources

Is the Relation a Function?

Try It: Is the Relation a Function?