1.4 Representing Functions Using Tables
A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
Table 1-3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]\text{}f\text{}[/latex] where [latex]\text{}D=f\left(m\right)\text{}[/latex] identifies months by an integer rather than by name.
| Month number, [latex]\text{}m\text{}[/latex] (input) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Days in month, [latex]\text{}D\text{}[/latex] (output) | 31 | 28 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 | 31 |
Table 1-4 defines a function [latex]\text{}Q=g\left(n\right)\text{}[/latex]. Remember, this notation tells us that [latex]\text{}g\text{}[/latex] is the name of the function that takes the input [latex]\text{}n\text{}[/latex] and gives the output [latex]\text{}Q\text{}[/latex].
| [latex]n[/latex] | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| [latex]Q[/latex] | 8 | 6 | 7 | 6 | 8 |
Table 1-5 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.
| Age in years, [latex]\text{ }a\text{ }[/latex] (input) | 5 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|
| Height in inches, [latex]\text{ }h\text{ }[/latex] (output) | 40 | 42 | 44 | 47 | 50 | 52 | 54 |
How To
Given a table of input and output values, determine whether the table represents a function.
- Identify the input and output values.
- Check to see if each input value is paired with only one output value. If so, the table represents a function.
Example 1: Identifying Tables that Represent Functions
Which table, Table 1-6, Table 1-7, or Table 1-8, represents a function (if any)?
| Input | Output |
|---|---|
| 2 | 1 |
| 5 | 3 |
| 8 | 6 |
| Input | Output |
|---|---|
| –3 | 5 |
| 0 | 1 |
| 4 | 5 |
| Input | Output |
|---|---|
| 1 | 0 |
| 5 | 2 |
| 5 | 4 |
Example 2: Identifying Functions
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