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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.

& 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

Does this set of ordered pairs represent a function?

Watch This

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 y for every value of x .

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

Table A
x y
0 4
1 7
2 7
3 6
Table B
x y
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.

x y
-3.5 -3.6
-1 -1
4 3.6
7.8 7.2

Solution:

The relation is a function because there is only one value of y for every value of x .

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

& 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

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 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 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.

s = \{(1, 2), (2, 2), (3, 2), (4, 2)\}

2. Which of the following relations represent a function? Explain.

3. Which of the following relations represent a function? Explain.

a) & 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
b)
c)

Answers:

1. s=\{(1,2),(2,2),(3,2),(4,2)\}

This is a function because there is one output for every input. In other words, if you think of these points as coordinate points (x, y) , there is only one value for y given for every value of x .

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) & 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

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. .
& X \qquad 2 \qquad \quad 3 \qquad 2 \qquad \quad \ 5\\& Y \qquad 3 \qquad -1 \qquad 5 \qquad -4
  1. .
& X \qquad 4 \qquad 2 \qquad \quad 6 \qquad -1\\& Y \qquad 2 \qquad 4 \qquad -3 \qquad \quad 5
  1. .
& X \qquad 1 \qquad 2 \qquad 3 \qquad 4\\& Y \qquad 5 \qquad 8 \qquad 5 \qquad 8
  1. .
& X \qquad -6 \qquad -5 \qquad -4 \qquad -3\\& Y \qquad \quad 4 \qquad \quad \ 4 \qquad \quad \ 4 \qquad \quad \ 4
  1. .
& X \qquad -2 \qquad 0 \qquad -2 \qquad 4\\& Y \qquad \quad 6 \qquad \ 4 \qquad \quad \ 4 \qquad 6

Which of the following relations represent a function? Explain.

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

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Difficulty Level:

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Date Created:

Jan 16, 2013

Last Modified:

Jul 15, 2014
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