Introduction
A function table records selected inputs and their corresponding outputs. To read its domain, focus on the input information: which input values or input tuples have an output recorded under the table’s stated convention?
This is a question about what the table represents, not about guessing what might happen at inputs that are absent. A finite table lists only finite information. Its listed domain consists of the inputs for which the table supplies a function value.
What Does “Listed Domain” Mean?
The domain represented by a table, or its listed domain, is the set of inputs for which that table gives a function value according to its stated conventions. For a one-input table, the inputs are individual values. For a two-input table, each input is an ordered pair.
The listed domain describes the information in the table. It does not automatically describe every input at which some larger function might be defined. A separate formula or statement could supply additional information, but a table by itself does not authorize us to fill in unlisted values.
A function table must also assign no more than one output to any one input. If the same input is listed with two different outputs and the table gives no rule resolving the conflict, the entries do not define a function as written.
Read the Table Before Listing the Domain
Before writing a set, identify the parts of the table:
- Input labels: These identify the values or coordinates supplied to the function.
- Output labels: These identify the function values recorded for those inputs.
- Legend or convention: This explains any special notation, such as a dash or a blank cell.
A blank cell has no universal meaning. One table may state that blank cells mean “no value is defined here.” Another may use them to mean “not recorded” or “information unavailable.” Those descriptions are not interchangeable. Report what the table’s stated convention supports; if no convention is supplied, do not silently assign one.
Likewise, a recorded output of \(0\) is still a value. It is not a blank entry, and it does not remove its input from the listed domain.
One-Input Tables
In a one-input table, each row usually pairs an input \(x\) with an output \(f(x)\). To find the listed domain, read the input column and include the inputs whose output cells contain values under the stated convention. Do not include an input just because it appears in a label or in a row with no represented output.
Worked Example: Reading a One-Input Table
Suppose the table uses the convention “a dash means that no function value is listed.” Its entries are:
| \(x\) | \(f(x)\) |
|---|---|
| \(-2\) | \(5\) |
| \(0\) | \(0\) |
| \(3\) | — |
| \(5\) | \(-1\) |
First read the convention: the dash does not represent an output. The row \(x=-2\) lists output \(5\), so \(-2\) is included. The row \(x=0\) lists output \(0\), so \(0\) is included. The row \(x=3\) has a dash, so \(3\) is not included. The row \(x=5\) lists output \(-1\), so \(5\) is included.
Thus the listed domain is
Check: Each element of the set matches a row with a recorded output: \(f(-2)=5\), \(f(0)=0\), and \(f(5)=-1\). The only other input shown is \(3\), whose dash is excluded by the stated convention.
Two-Input Tables and Ordered Pairs
For a function of two variables, an input has two coordinates. A grid table often puts one coordinate in the row headings and the other in the column headings. The input associated with a cell is the ordered pair formed from those two headings.
For example, if a cell lies in the row labeled \(y=4\) and the column labeled \(x=-1\), its input is \((-1,4)\), in the order \((x,y)\). The output displayed in that cell, if the convention counts it as a value, is \(f(-1,4)\).
Worked Example: Reading a Two-Input Grid
Suppose a grid uses columns for \(x\), rows for \(y\), and the legend says “a blank cell means no value is listed.” The table is:
| \(y\backslash x\) | \(-1\) | \(2\) | \(4\) |
|---|---|---|---|
| \(0\) | \(7\) | \(3\) | |
| \(3\) | \(0\) | \(8\) |
Read each cell that contains an output. In the row \(y=0\), the cell under \(x=-1\) gives input \((-1,0)\), and the cell under \(x=4\) gives input \((4,0)\). The cell under \(x=2\) is blank and is excluded. In the row \(y=3\), the cell under \(x=2\) gives \((2,3)\), and the cell under \(x=4\) gives \((4,3)\). The blank cell under \(x=-1\) is excluded.
Therefore, the listed domain is
The entry \(0\) at \((2,3)\) is an output, so that pair belongs in the domain. There are \(2\) rows and \(3\) columns, giving \(2\cdot 3=6\) possible grid positions. Two are blank, leaving \(6-2=4\) listed inputs, exactly the four pairs above.
Check: The set contains precisely the four coordinate pairs corresponding to occupied cells. Each pair is written in the order \((x,y)\), even though \(y\) labels the rows.
Marked Entries Depend on the Legend
A symbol such as a dash, an asterisk, or “N/A” can be used in different ways. Its meaning must come from the table’s convention. If the legend says a mark means “undefined,” then the marked input is not in the listed domain. If it says the mark means “not measured,” then the table does not provide a function value there; do not convert that mark into a numerical output. If a mark is assigned some other explicit meaning, follow that meaning.
Worked Example: Applying a Marked-Entry Convention
A table lists input pairs \((u,v)\) directly. Its legend says “an asterisk means the output is not recorded; only numerical output entries count as listed function values.” Find the listed domain.
| \(u\) | \(v\) | Output |
|---|---|---|
| \(1\) | \(-2\) | \(6\) |
| \(1\) | \(0\) | * |
| \(3\) | \(-2\) | \(0\) |
| \(3\) | \(0\) | \(4\) |
The first row has numerical output \(6\), so it contributes \((1,-2)\). The second row has an asterisk, and the legend says that this is not a recorded output, so it contributes no input to the listed domain. The third row has output \(0\), which is numerical and therefore contributes \((3,-2)\). The fourth row has output \(4\), so it contributes \((3,0)\).
The listed domain is
Check: There are four input rows in total. Exactly one output is marked with an asterisk, so \(4-1=3\) rows contribute inputs. The set has three ordered pairs and includes the row with output \(0\).
Why This Reading Rule Works
The rule follows directly from what it means for a table to represent a function value. An input belongs to the listed domain precisely when the table gives an output for that input. For a two-input table, the “input” is the ordered pair, not either coordinate on its own.
If the convention identifies an entry as an output, the row or cell associates that output with a particular input.
By definition, an input with a represented function value belongs to the listed domain. In a grid, use the ordered pair of the column and row labels in the stated coordinate order.
If the convention says a blank or mark supplies no function value, that cell supplies no input to the listed domain.
Every represented value appears in one of the table’s entries. An input absent from the table, or paired only with an entry that the convention excludes, has no listed value in that table.
These steps establish both directions: every input paired with a represented output is included, and every input included in the listed domain must be paired with a represented output. Therefore the set obtained by reading those entries is exactly the listed domain.
A Common Mistake: Filling in the Whole Grid
A frequent error is to assume that every combination of displayed row and column labels belongs to the domain. That is true only if every corresponding cell is represented as an output. A grid can display possible coordinate labels while leaving some combinations without a listed value.
In the previous grid, the row labels were \(0\) and \(3\), and the column labels were \(-1\), \(2\), and \(4\). One incorrect calculation is to take every combination and report
This set has \(2\cdot 3=6\) pairs, but it incorrectly includes \((2,0)\) and \((-1,3)\), whose cells were blank. The correction is to retain only pairs whose cells contain outputs under the stated convention:
The same principle prevents another mistake: treating a blank as \(0\). In that grid, \(0\) was explicitly displayed at \((2,3)\), so that input is included. A blank cell has no displayed \(0\); it is handled according to the legend.
Reporting the Result Clearly
When writing a listed domain, use a set and preserve the input structure. For one input, list values such as \(\{-2,0,5\}\). For two inputs, list ordered pairs such as \(\{(-1,0),(4,0)\}\). If the table contains no represented outputs, the listed domain is the empty set, written \(\varnothing\).
It can also help to state the convention in words before giving the set. For example: “Blank cells mean no value is listed, so the listed domain is the set of coordinate pairs corresponding to nonblank cells.” That makes the treatment of missing entries explicit.
Finally, call the result the listed domain when you mean what this finite table records. Do not claim that no other inputs are possible for a function beyond the information represented by the table.
Check Your Understanding
Answer each question using only the entries and conventions stated.
- A one-input table lists \(x=2\) with output \(0\). Does \(2\) belong to the listed domain? Explain.
- A grid has columns \(x=-1,1\) and rows \(y=0,2\). The cell at \(x=1,y=2\) contains \(5\). What input pair does that entry represent, using the order \((x,y)\)?
- A table’s legend says that a dash means “no value is listed.” Should an input paired with a dash be included in the listed domain?
- Why is it incorrect to include every combination of row and column labels when some cells are blank?
- If a table gives no meaning for its blank cells, what should you do before deciding whether those cells count as outputs?