3.1: Relations vs Functions
Is There a Difference Between a Relation and a Function?
Objectives
The lesson objective for Relations vs Functions is:
 Is there a difference between a relation and a function?
Introduction
In this introductory concept for chapter A Picture is Worth a Thousand Words, you will be introduced to two new terms, a relation and a function. It is important for you to keep these two terms clear in your mind. If you think of the relationship between two quantities, 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 have more than one output for every input. 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*}
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.
Watch This
Khan Academy Functions as Graphs
Guidance
It is often said that all functions are relations but not all relations are functions. How can this be true?
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*}
If your look at a set of ordered pairs (relation) and there is only one value of \begin{align*}y\end{align*}
On the other hand if the relation shows that there is more than one output \begin{align*}(y)\end{align*}
Look at the two tables below. Table below shows a relation that is a function. Table below 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 
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 
The relation is a function because there is only one value of \begin{align*}y\end{align*}
Example B
Which one of the following graphs represents a function?
In order to answer this question, you need to know about the 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.
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?
Vocabulary
 Function

A function is a set of ordered pairs \begin{align*}(x, y)\end{align*}
(x,y) 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*}x , there is only one value for \begin{align*}y\end{align*}y .
 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.
Summary
Knowing the difference between a relation and a function is an important step in learning about linear and nonlinear equations. A function is a relationship between quantities where there is one output for every input. If you have more than one output for a particular input, then the quantities represent a relation. A graph of a relationship can be shown to be a function using the vertical line test. If the 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.
Problem Set
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*}
Answers
Determine whether each of the following...
 This graph represents a function. It passes the vertical line test. The vertical line crosses the graph in one place only.
 This graph does not represent a function. It does not pass the vertical line test. The vertical line crosses the graph in more than one place.
 This graph represents a function. It passes the vertical line test. The vertical line crosses the graph in one place only.
 This graph does not represent a function. It does not pass the vertical line test. The vertical line crosses the graph in more than one place.
 This graph does not represent a function. It does not pass the vertical line test. The vertical line crosses the graph in more than one place.
Which of the following...
 This table of values does not represent a function because for every input there is more than one output. The \begin{align*}x\end{align*}value of 2 produces two distinct \begin{align*}y\end{align*}values.
 This table of values does represent a function because for every input there is only one output. Each \begin{align*}x\end{align*}value results in one \begin{align*}y\end{align*}value.
 This table of values does represent a function because for every input there is only one output. Each \begin{align*}x\end{align*}value results in one \begin{align*}y\end{align*}value.
 This table of values does represent a function because for every input there is only one output. Each \begin{align*}x\end{align*}value results in one \begin{align*}y\end{align*}value.
 This table of values does not represent a function because for every input there is more than one output. The \begin{align*}x\end{align*}value of 2 produced two distinct \begin{align*}y\end{align*}values.
Which of the following...
 \begin{align*}s = \{(1, 1), (1, 2), (1,3), (1, 4), (1, 5)\}\end{align*} This set of values does not represent a function because for every input there is more than one output. The \begin{align*}x\end{align*}value 1 produces five different \begin{align*}y\end{align*}values.
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