3.2: Relations and Functions
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?
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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. Table B shows a relation that is not a function.
\begin{align*}x\end{align*}  \begin{align*}y\end{align*} 

0  0 
1  1 
2  2 
3  3 
\begin{align*}x\end{align*}  \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.
\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 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
\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 a relation.
Vocabulary
 Function
 A function is a set of ordered pairs \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 \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 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.
 \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)
 d)
Answers:
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)
 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.

 \begin{align*}& X \qquad 2 \qquad \quad 3 \qquad 2 \qquad \quad \ 5\\ & Y \qquad 3 \qquad 1 \qquad 5 \qquad 4\end{align*}

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

 \begin{align*}& X \qquad 1 \qquad 2 \qquad 3 \qquad 4\\ & Y \qquad 5 \qquad 8 \qquad 5 \qquad 8\end{align*}

 \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*}

 \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 does NOT represent a function? Explain.
 \begin{align*}s=\{(3,3),(2,2),(1,1),(0,0),(1,1),(2,2),(3,3)\}\end{align*}
 \begin{align*}s=\{(1,1),(1,2),(1,3),(1,4),(1,5)\}\end{align*}
 \begin{align*}s=\{(1,1),(2,1),(3,1),(4,1),(5,1)\}\end{align*}
 \begin{align*}s=\{(3,9),(2,4),(1,1),(1,1),(2,4)\}\end{align*}
 \begin{align*}s=\{(3,3),(2,2),(1,1),(0,0),(1,1),(2,2)\}\end{align*}
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Term  Definition 

Function  A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for . 
Relation  A relation is any set of ordered pairs . 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. 
Image Attributions
Here you will learn about relations, and what makes a relation a function.