3.2: Relations and Functions
The following table of values represents data collected by a student in a math class.
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 for every value of .
Look at the two tables below. Table A shows a relation that is a function because every value has only one value. Table B shows a relation that is not a function because there are two different values for the value of 0.
0 | 4 |
1 | 7 |
2 | 7 |
3 | 6 |
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.
4 | 3.6 |
7.8 | 7.2 |
Solution:
The relation is a function because there is only one value of for every value of .
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
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 , there is only one value for .
- Relation
- A relation is any set of ordered pairs . 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.
2. Which of the following relations represent a function? Explain.
3. Which of the following relations represent a function? Explain.
- a)
- b)
- c)
Answers:
1.
- This is a function because there is one output for every input. In other words, if you think of these points as coordinate points , there is only one value for given for every value of .
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)
- 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.
- .
- .
- .
- .
- .
Which of the following relations represent a function? Explain.
Image Attributions
Description
Learning Objectives
Here you will learn about relations, and what makes a relation a function.