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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.

\begin{align*}& x \qquad 5 \qquad \ 10 \qquad 15 \qquad 10 \qquad \ 5 \qquad \ 0\\ & y \qquad 12 \qquad 25 \qquad 37 \qquad 55 \qquad 72 \qquad 0\end{align*}

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 \begin{align*}y\end{align*} for every value of \begin{align*}x\end{align*}.

Look at the two tables below. Table A shows a relation that is a function because every \begin{align*}x\end{align*} value has only one \begin{align*}y\end{align*} value. Table B shows a relation that is not a function because there are two different \begin{align*}y\end{align*} values for the \begin{align*}x\end{align*} value of 0.

Table A
\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 4
1 7
2 7
3 6
Table B
\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 4
0 2
2 6
2 7

When looking at the graph of a relation, you can determine whether or not it is a function using the vertical line test. If a vertical line can be drawn anywhere through the graph such that it intersects the graph more than once, the graph is not function.

#### Example A

Determine if the following relation is a function.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
\begin{align*}-3.5\end{align*} \begin{align*}-3.6\end{align*}
\begin{align*}-1\end{align*} \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 \begin{align*}y\end{align*} for every value of \begin{align*}x\end{align*}.

#### Example B

Which of the following graphs represent 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 of the following represent functions?

Solution:

a) This is a function because every input has only one output.

b) This is not a function because one input (1) has two outputs (2 and 7).

c) This is a function because every input has only one output.

#### Concept Problem Revisited

\begin{align*}& x \qquad 5 \qquad \ 10 \qquad 15 \qquad 10 \qquad \ 5 \qquad \ 0\\ & y \qquad 12 \qquad 25 \qquad 37 \qquad 55 \qquad 72 \qquad 0\end{align*}

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 just a relation.

### Vocabulary

Function
A function is an example of a relation where there is only one output for every input. In other words, for every value of \begin{align*}x\end{align*}, there is only one value for \begin{align*}y\end{align*}.
Relation
A relation is any set of ordered pairs \begin{align*}(x, y)\end{align*}. A relation can have more than one output for an input.
Vertical Line Test
The Vertical Line Test is a test for functions. If you can take your pencil and draw a straight vertical line through any part of the graph, and the pencil hits the graph more than once, the graph is not a function.

### Guided Practice

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

\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) \begin{align*}& x \qquad 2 \qquad 4 \qquad \ 6 \qquad \ 8 \qquad \ 10 \qquad 12\\ & y \qquad 3 \qquad 7 \qquad 11 \qquad 15 \qquad 19 \qquad 23\end{align*}
b)
c)

1. \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 \begin{align*}(x, y)\end{align*}, there is only one value for \begin{align*}y\end{align*} given for every value of \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) \begin{align*}& x \qquad 2 \qquad 4 \qquad \ 6 \qquad \ 8 \qquad \ 10 \qquad 12\\ & y \qquad 3 \qquad 7 \qquad 11 \qquad 15 \qquad 19 \qquad 23\end{align*}

This is a function because there is only one output for a given input.
b)
Since the vertical line hit the graph more than once (indicated by the three blue circles), the graph does not represent a function.
c)
Since the vertical line hit the graph only once (indicated by the one blue dot), the graph does represent a function.

### Practice

Determine whether or not each relation is a function. Explain your reasoning.

1. .

1. .

1. .

1. .

1. .

Which of the following relations represent a function? Explain.

1. .
\begin{align*}& X \qquad 2 \qquad \quad 3 \qquad 2 \qquad \quad \ 5\\ & Y \qquad 3 \qquad -1 \qquad 5 \qquad -4\end{align*}
1. .
\begin{align*}& X \qquad 4 \qquad 2 \qquad \quad 6 \qquad -1\\ & Y \qquad 2 \qquad 4 \qquad -3 \qquad \quad 5\end{align*}
1. .
\begin{align*}& X \qquad 1 \qquad 2 \qquad 3 \qquad 4\\ & Y \qquad 5 \qquad 8 \qquad 5 \qquad 8\end{align*}
1. .
\begin{align*}& X \qquad -6 \qquad -5 \qquad -4 \qquad -3\\ & Y \qquad \quad 4 \qquad \quad \ 4 \qquad \quad \ 4 \qquad \quad \ 4\end{align*}
1. .
\begin{align*}& X \qquad -2 \qquad 0 \qquad -2 \qquad 4\\ & Y \qquad \quad 6 \qquad \ 4 \qquad \quad \ 4 \qquad 6\end{align*}

Which of the following relations represent a function? Explain.

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

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English

TermDefinition
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 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 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.

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