Introduction

A function is a rule where every input value is assigned precisely one output value.

As an example, let's consider the rule defined by

f(x) = x+3

where x can be any number from the set \{1,2,3,4\}. A set is a collection of numbers we can use in our rule.

Let's calculate the possible values of f(x)\mathbin{:}

\begin{align*} f(1) &= 1+3 = 4\\[5pt] f(2) &= 2+3 = 5\\[5pt] f(3) &= 3+3 = 6\\[5pt] f(4) &= 4+3 = 7 \end{align*}

We can represent this rule using a mapping diagram, like the one shown below.

This rule is a function because every input is mapped to precisely one output.

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Examples of Rules That Are Not Functions

Now let's consider the rule given by the following mapping diagram:

This rule is not a function because at least one input is assigned to more than one output. In this case, the input x=1 is mapped to both 2 and -2.

Finally, let's consider the rule given by the following mapping diagram:

This rule is not a function because at least one input is not assigned to an output. In this case, the input 0 is not assigned an output.

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A Summary

So, we must remember that a rule is a function if and only if every input is mapped to precisely one output.

In particular, a rule is not a function in the following cases:

  • An input value is not assigned an output value. For example, the following situation is not allowed for functions (an input value with no arrow from it):
  • An input value is assigned to more than one output value. For example, the following situation is not allowed for functions (an input value with more than one arrow from it):
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Example: Determining if a Rule Given by a Mapping Diagram Is a Function

Is the following rule a function?

EXPLANATION

A rule is a function if and only if every input is mapped to precisely one output.

In particular, a rule is not a function in the following cases:

  • An input value is not assigned to an output value. For example, the following situation is not allowed for functions (an input value with no arrow from it):
  • An input value is assigned to more than one output value. For example, the following situation is not allowed for functions (an input value with more than one arrow from it):

In this particular example, the rule is a function because each input value is assigned precisely one output value.

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Practice: Determining if a Rule Given by a Mapping Diagram Is a Function

Is the above rule a function?

a
 Yes, since both sets have the same number of elements
b
 No, since both sets have the same number of elements
c
 Not enough information
d
 Yes, since each input is assigned to exactly one output
e
 Yes, since each output is related to at least one input
Practice: Determining if a Rule Given by a Mapping Diagram Is a Function

Is the above rule a function?

a
 No, since both sets have different numbers of elements
b
 No, since there is at least one output with more than one input
c
 Yes, since each input is assigned to exactly one output
d
 Yes, since each output is related to at least one input
e
 Not enough information
Rules Represented Using Tables

When a rule is represented as a table, we have a function if and only if all values in the first row are distinct (i.e., different).

Let's consider some examples.

The rule defined by the following table is a function. Notice that all values in the first row are distinct. Every input value is assigned to exactly one output value.

x 1 2 3 4 5 6
f(x) -2 2 0 7 3 0

On the other hand, the rule defined by the following table is not a function. Notice that the first row contains the value \color{red}3 twice. The input value 3 is assigned to two output values ( 0 and -1 ).

x 1 2 \color{red}3 4 \color{red}3 6
f(x) -2 2 0 7 -1 5
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Example: Determining if a Rule Given by a Table Is a Function

Which missing xy -pair would make the following rule a function?

x -3 -1 ? 3 5 7 9
y 0 4 ? 0 2 1 7
  1. (1,2)
  2. (7,3)
  3. (-1,5)
EXPLANATION

A rule is a function if and only if every input is mapped to precisely one output.

  • If we substitute the xy -pair (1,2) into our table, we get a function.

    x -3 -1 \color{blue}1 3 5 7 9
    y 0 4 2 0 2 1 7

    Notice that all values in the first row are distinct (i.e., different). Every input value is assigned to precisely one output value.

  • None of the other xy -pairs gives rules that are functions. So, for example, substituting the xy -pair (-1,5) gives a rule that is not a function.

    x -3 \color{red}-1 \color{red}-1 3 5 7 9
    y 0 4 5 0 2 1 7

    Notice that the first row contains the value \color{red}x=-1 twice. The input value -1 is assigned to more than one output value ( 4 and 5 ).

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Practice: Determining if a Rule Given by a Table Is a Function

Which missing $xy$-pair would make the following rule a function?

$x$ $-5$ ? $-1$ $0$ $1$ $3$ $5$
$y$ $25$ ? $1$ $0$ $2$ $4$ $6$
a
 $(0, -1)$
b
 $(-1, -4)$
c
 $(-5, -10)$
d
 $(-2, -4)$
e
 $(3, 5)$
Practice: Determining if a Rule Given by a Table Is a Function

Consider a rule between $x$ and $y$ given by the table below.

$x$ $12$ $9$ $7$ $3$ $6$ $9$
$y$ $15$ $12$ $9$ $6$ $3$ $0$
\[\] Which of the following statements are true?

  1. The rule is a function
  2. The rule is not a function
  3. The rule maps the input $x=15$ to the output $y=12$
a
 III only
b
 I and III only
c
 I and II only
d
 II only
e
 I only
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