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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Khan Academy Functions as Graphs
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.
\begin{align*}x\end{align*}  \begin{align*}y\end{align*} 

0  4 
1  7 
2  7 
3  6 
\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)
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)
 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.
 .
 .
 .
 .
 .
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 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*}
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.
Concept Nodes:
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.