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# 3.2: Relations and Functions

Difficulty Level: Advanced Created by: CK-12
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The following table of values represents data collected by a student in a math class.

x5 101510 5 0y12253755720

Does this set of ordered pairs represent a function?

### Guidance

Consider the relationship between two variables. You can think of this relationship in terms of an input/output machine.

If there is only one output for every input, you have a function. If not, you have a relation. Relations can have more than one output for every input.

A relation is any set of ordered pairs. A function is a set of ordered pairs where there is only one value of y\begin{align*}y\end{align*} for every value of x\begin{align*}x\end{align*}.

Look at the two tables below. Table A shows a relation that is a function. Table B shows a relation that is not a function.

Table A
x\begin{align*}x\end{align*} y\begin{align*}y\end{align*}
0 0
1 1
2 2
3 3
Table B
x\begin{align*}x\end{align*} y\begin{align*}y\end{align*}
0 0
1 1
2 2
2 1

A graph of a relation can be shown to be a function using the vertical line test. If a vertical line can be drawn through the graph such that it intersects the graph line more than once, the graph is not function but a relation.

#### Example A

Determine if the following relation is a function.

x\begin{align*}x\end{align*} y\begin{align*}y\end{align*}
3.5\begin{align*}-3.5\end{align*} 3.6\begin{align*}-3.6\end{align*}
1\begin{align*}-1\end{align*} 1\begin{align*}-1\end{align*}
4 3.6
7.8 7.2

Solution:

The relation is a function because there is only one value of y\begin{align*}y\end{align*} for every value of x\begin{align*}x\end{align*}.

#### Example B

Which one of the following graphs represents a function?

Solution:

In order to answer this question, you need to use the vertical line test. A graph represents a function if no vertical line intersects the graph more than once. Let’s look at the first graph. Draw a vertical line through the graph.

Since the vertical line hit the graph more than once (indicated by the two red dots), the graph does not represent a function.

Since the vertical line hit the graph only once (indicated by the one red dot), the graph does represent a function.

Since the vertical line hit the graph only once (indicated by the one red dot), the graph does represent a function.

Since the vertical line hit the graph more than once (indicated by the three red dots), the graph does not represent a function.

#### Example C

Which one of the following represents a function?

Solution:

#### Concept Problem Revisited

x5 101510 5 0y12253755720

If you look at this table, there are two places where you see the more than one output for a single input.

You can conclude that this set of ordered pairs does not represent a function. It is a relation.

### Vocabulary

Function
A function is a set of ordered pairs (x,y)\begin{align*}(x, y)\end{align*} that shows a relationship where there is only one output for every input. In other words, for every value of x\begin{align*}x\end{align*}, there is only one value for y\begin{align*}y\end{align*}.
Relation
A relation is any set of ordered pairs (x,y)\begin{align*}(x, y)\end{align*}. A relation has more than one output for an input.
Vertical Line Test
The Vertical Line Test is a test for functions. If you take your pencil and draw a straight line through any part of the graph, and the pencil hits the graph more than once, the graph is not a function. Therefore, a graph will represent a function if the vertical line test passes, In other words, no vertical line intersects the graph more than once.

### Guided Practice

1. Is the following a representation of a function? Explain.

s={(1,2),(2,2),(3,2),(4,2)}\begin{align*}s = \{(1, 2), (2, 2), (3, 2), (4, 2)\}\end{align*}

2. Which of the following relations represent a function? Explain.

3. Which of the following relations represent a function? Explain.

a)
x24 6 8 1012y3711151923
b)
c)
d)

1. s={(1,2),(2,2),(3,2),(4,2)}\begin{align*}s=\{(1,2),(2,2),(3,2),(4,2)\}\end{align*}

This is a function because there is one output for every input. In other words, if you think of these points as coordinate points (x,y)\begin{align*}(x, y)\end{align*}, there is only one value for y\begin{align*}y\end{align*} given for every value of x\begin{align*}x\end{align*}.

2. a)

Since the vertical line hit the graph more than once (indicated by the two green circles), the graph does not represent a function.
b)
Since the vertical line hit the graph only once (indicated by the one green dot), the graph does represent a function.

3. a)

x24 6 8 1012y3711151923

This is a function because there is only one output for a given input.
b)
This is not a function because there is more than one output for a given input. For the input number 2, there are two output values (7 and 9)
c)
Since the vertical line hit the graph more than once (indicated by the three blue circles), the graph does not represent a function.
d)
Since the vertical line hit the graph only once (indicated by the one blue dot), the graph does represent a function.

### Practice

Determine whether each of the following is a relation or a function. Explain your reasoning.

Which of the following relations represent a function? Explain.

X232 5Y3154

X4261Y2435

X1234Y5858

X6543Y4 4 4 4

X2024Y6 4 46

Which of the following relations does NOT represent a function? Explain.

1. s={(3,3),(2,2),(1,1),(0,0),(1,1),(2,2),(3,3)}\begin{align*}s=\{(-3,3),(-2,-2),(-1,-1),(0,0),(1,1),(2,2),(3,3)\}\end{align*}
2. s={(1,1),(1,2),(1,3),(1,4),(1,5)}\begin{align*}s=\{(1,1),(1,2),(1,3),(1,4),(1,5)\}\end{align*}
3. s={(1,1),(2,1),(3,1),(4,1),(5,1)}\begin{align*}s=\{(1,1),(2,1),(3,1),(4,1),(5,1)\}\end{align*}
4. s={(3,9),(2,4),(1,1),(1,1),(2,4)}\begin{align*}s=\{(-3,9),(-2,4),(-1,1),(1,1),(2,4)\}\end{align*}
5. s={(3,3),(2,2),(1,1),(0,0),(1,1),(2,2)}\begin{align*}s=\{(3,-3),(2,-2),(1,-1),(0,0),(-1,1),(-2,2)\}\end{align*}

### Vocabulary Language: English

Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Relation

Relation

A relation is any set of ordered pairs $(x, y)$. A relation can have more than one output for a given input.
Vertical Line Test

Vertical Line Test

The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function.

Jan 16, 2013

Feb 26, 2015