Reading a Complete Function Table
A table records which output is paired with each input. When the table gives the complete list of inputs for a finite function, its domain and range can be read directly from those pairs. The main care needed is to treat them as sets: a value that appears more than once is listed only once.
This is a different task from finding a rule or extending a function beyond the entries shown. We use only the input-output pairs in the table. If the table is not stated to be complete, it may show just some values of a function, and we cannot assume that it gives the function's entire domain or range.
Read the Columns as Input-Output Pairs
Suppose a table has an input column labelled \(x\) and an output column labelled \(f(x)\). Each row gives one ordered pair \((x,f(x))\). The domain is made from the first coordinates of the pairs, and the range is made from the second coordinates.
The table’s row order does not matter. A table can list inputs in increasing order, decreasing order, or any other order: the same input-output pairs give the same domain and range. Nor does an output have to occur only once. Different inputs can produce the same output, and that output still appears only once in the range as a set.
For instance, if \(f(1)=4\) and \(f(3)=4\), both inputs belong to the domain, but the output \(4\) is listed once in the range. The range records which values are attained, not how many times each value occurs.
Worked Example: Find Both Sets from an Unordered Table
Worked Example: Find Both Sets from an Unordered Table
Find the domain and range of the function represented by this complete table.
| \(x\) | \(f(x)\) |
|---|---|
| \(3\) | \(7\) |
| \(-2\) | \(7\) |
| \(5\) | \(-4\) |
| \(0\) | \(1\) |
First collect the inputs from the left column: \(3,-2,5,0\). Since the domain is a set, we can write them in a convenient order:
Next collect the outputs from the right column: \(7,7,-4,1\). The value \(7\) appears twice, but a set lists it only once. Thus:
Check against the rows: the four inputs \(-2,0,3,5\) are exactly the inputs shown. The outputs shown are \(7,7,-4,1\), whose distinct values are \(-4,1,7\). Both sets include every value from the relevant column and no others.
Why Repeated Values Appear Only Once
A set is determined by its members, not by how many times those members are written. Therefore \(\{2,2,5\}\) and \(\{2,5\}\) name the same set. This matters for range: the table may have several rows with the same output, but those rows do not create several different output values.
We can justify the extraction method directly. Let a complete table have rows \((x_1,y_1),\ldots,(x_n,y_n)\), where \(y_i=f(x_i)\). The definition of domain says that a value belongs to the domain exactly when it occurs as an input in one of these rows. Hence the domain is \(\{x_1,\ldots,x_n\}\). The definition of range says that a value belongs to the range exactly when it is the output for at least one input in the domain. Every such output is one of \(y_1,\ldots,y_n\), and each listed \(y_i\) is an output of its corresponding input. So the range is \(\{y_1,\ldots,y_n\}\). Repeated entries do not change either set, because membership asks whether a value occurs, not how many times it occurs.
This argument also explains why row order is irrelevant: rearranging the rows changes the order in which values are encountered, but not which values occur.
Worked Example: Keep Repeated Outputs Once
Worked Example: Keep Repeated Outputs Once
Find the domain and range from the table.
| \(x\) | \(g(x)\) |
|---|---|
| \(4\) | \(3\) |
| \(-1\) | \(6\) |
| \(2\) | \(3\) |
| \(0\) | \(-2\) |
The input entries are \(4,-1,2,0\). Put them in increasing order to write the domain:
The output entries are \(3,6,3,-2\). The value \(3\) occurs for two different inputs, but the range contains distinct output values, not a count of output occurrences:
Check by reading the table again. Every input appears in the domain, and there are no additional inputs. Each range value occurs in the output column: \(-2\) occurs once, \(3\) occurs twice, and \(6\) occurs once. Removing the repeated \(3\) leaves exactly \(\{-2,3,6\}\).
A Table Must Agree with the Function Rule
A function assigns exactly one output to each input in its domain. Accordingly, if an input appears more than once in a table, its output must be the same in every row containing that input. Repeating the same pair does not alter the domain or range. But assigning two different outputs to the same input would make the table inconsistent with the function rule.
Worked Example: Repeated Input, Consistent Output
Worked Example: Repeated Input, Consistent Output
Determine the domain and range of the function represented by this complete table, and check whether the repeated input causes a problem.
| \(x\) | \(h(x)\) |
|---|---|
| \(-1\) | \(6\) |
| \(4\) | \(3\) |
| \(0\) | \(-2\) |
| \(-1\) | \(6\) |
The input values are \(-1,4,0,-1\). The repeated \(-1\) is included once, so:
The output values are \(6,3,-2,6\). Removing the repeated \(6\) gives:
The repeated input \(-1\) is paired with \(6\) in both rows, so the table is consistent with the function rule. Check the sets against the rows: the distinct inputs are exactly \(-1,0,4\), and the distinct outputs are exactly \(-2,3,6\).
Common Mistake: Counting Occurrences Instead of Values
A frequent error is to write an output in the range as many times as it appears in the table. For the second worked example, an incorrect calculation would be:
This lists the rows’ outputs, but it is not the range as a set in simplified form: the repeated \(3\) is the same value, not a new range member. The correction is:
The same set principle applies to the domain if an input happens to be repeated. Write each distinct input once. Do not, however, use this principle to overlook a function-rule problem: if a repeated input is paired with different outputs, the table does not represent a function.
What the Table Does and Does Not Tell You
For a complete table of a finite function, the rows give the entire domain and the attained outputs. The range is not necessarily the same as every output that might be allowed in some larger set. For example, if a function is described as taking values in the real numbers, the table’s range still consists only of the real values that actually appear as outputs in its rows.
Also pay attention to whether the table is complete. If it is only a sample of values, it tells you the inputs and outputs shown, but it may not reveal the function’s full domain or range. The extraction procedure in this tutorial applies when the table represents the complete finite function, as stated in the problem.
In short, read each row as a pair, collect the first coordinates for the domain and the second coordinates for the range, and remove repeats from both sets. This method is useful whenever a finite function is presented in tabular form, including when the rows are unordered or several inputs share an output.
Check Your Understanding
Use the complete table below for Questions 1–3.
| \(x\) | \(p(x)\) |
|---|---|
| \(2\) | \(-1\) |
| \(-3\) | \(4\) |
| \(5\) | \(-1\) |
| \(0\) | \(2\) |
- What is the domain of \(p\)?
- What is the range of \(p\)?
- Why does the range contain \(-1\) only once?
- If an input is repeated in a table, what must be true for the table to represent a function?
- Why can an incomplete sample table fail to determine the full domain and range?
