Domain and Range of a Relation: Pairs, Graphs and Formulas
The domain of a relation is its set of x-values; the range is its set of y-values. For ordered pairs or a table, collect each coordinate separately and list each distinct value once. For a graph, include only values reached by actual points—not every value between the extremes.
Quick answer: The domain of a relation is the set of all first values (x-values) and the range is the set of all second values (y-values). For the ordered pairs (-10, 3), (3, 2), (0, -8), (8, -4), (7, 6), the domain is {-10, 0, 3, 7, 8} and the range is {-8, -4, 2, 3, 6}. List each value once, even if it repeats.
How to find domain and range from ordered pairs
In (x, y), the first coordinate contributes to the domain and the second contributes to the range. Separate the columns if that makes the sorting easier. Repeated values appear only once in the answer.
Existing example 1
S = {(4, 6), (5, 7), (6, 8), (7, 6)}.
Domain: {4, 5, 6, 7}.
Range: {6, 7, 8}.
The y-value 6 appears twice in the pairs, but it contributes only one member to the range.
Existing example 2
T = {(−2, 0), (0, −1), (2, 1), (3, 0)}.
Domain: {−2, 0, 2, 3}.
Range: {−1, 0, 1}.
Zero can belong to both sets. Keep negative signs attached to their coordinates.
A relation need not be a function. For example, {(1, 2), (1, 5)} has domain {1} and range {2, 5}, even though the input 1 has two outputs.
How to read domain and range from a graph
Look horizontally for x-values reached by the graph, then vertically for y-values. Check separate pieces, gaps, holes, endpoint dots and arrows. A graph that continues indefinitely may have no leftmost, rightmost, highest or lowest point.
The axis breaks mark skipped values before this labelled coordinate window for S. The points are not joined by segments. Its domain is {4, 5, 6, 7}, not [4, 7]; its range is {6, 7, 8}, not [6, 8]. Values such as x = 4.5 do not occur.
An open endpoint excludes that point. Its x- or y-value may still belong if another point elsewhere on the graph has that same coordinate. Check the whole relation before excluding a value.
How to find domain and range from a formula
For real-valued formulas, avoid zero denominators, require nonnegative quantities inside even roots, and require positive logarithm arguments. Find the outputs separately; a domain restriction is not automatically the range.
Square-root example
For y = √(x − 3), x − 3 ≥ 0, so the domain is [3, ∞). The square root produces nonnegative outputs, and every such output is possible, so the range is [0, ∞). At x = 3, y = 0, so both finite endpoints are included.
Reciprocal example
For y = 1/(x + 2), x cannot equal −2. The domain is (−∞, −2) ∪ (−2, ∞). The output cannot equal 0; every other real output is possible because x = 1/y − 2. The range is (−∞, 0) ∪ (0, ∞).
Also follow any stated domain and real-world context. A time variable might include zero, and a count might allow only whole numbers. For a composition f(g(x)), x must be allowed by g and g(x) must be allowed by f.
State the answer in the right notation
Use braces for a finite set, such as {4, 5, 6, 7}. Use interval notation for a continuous interval: [a, b] includes both endpoints, while (a, b) excludes them. Infinity always takes a parenthesis. The symbol ∪ joins separate intervals.
For reference, OpenStax explains relations and coordinate sets and domain and range for graphs and formulas.
Solved: the question types you will see
Homework and test platforms ask about domain and range in a few fixed formats. Here is how to answer each one.
1. “What is the domain of this relation?” (ordered pairs)
Write every x-value once, in order. Example: {(-10, 3), (3, 2), (0, -8), (8, -4), (7, 6)} has domain {-10, 0, 3, 7, 8}.
2. “What is the range of this relation?”
Write every y-value once. For the same pairs the range is {-8, -4, 2, 3, 6}. If two pairs share a y-value, such as (1, 4) and (5, 4), list 4 only once.
3. “Identify the domain and range” from a table
The top (or left) row is x, the other row is y. For x: -5, -2, 1, 5 and y: 10, 11, 4, 10, the domain is {-5, -2, 1, 5} and the range is {4, 10, 11}.
4. From a graph or mapping diagram
For points, read each point’s coordinates. For a mapping diagram, the left oval is the domain and the right oval is the range. For a continuous graph, read the leftmost and rightmost x-values for the domain and the lowest and highest y-values for the range, and use interval notation such as [-3, 4].
5. “Is this relation a function?”
It is a function only if no x-value repeats with a different y-value. (2, 5) and (2, 7) together mean it is not a function; (2, 5) and (3, 5) are fine.
Check your understanding
Find the domain and range of R = {(−3, 2), (1, 2), (1, 5), (4, −1)}. Is it a function?
Show the coordinate sets and function check
Domain: {−3, 1, 4}. Range: {−1, 2, 5}. It is not a function because x = 1 has two different outputs, 2 and 5. Repeating y = 2 does not cause that failure.
Continue with Algebra 2 review
Use the Algebra 2 Online Center for related functions and graph review. If you are considering a paid guide, compare the existing Algebra II book’s preview and current format before choosing it.
Existing video explanation
Related to This Article
More math articles
- Quotient Quickies: How to Navigate Decimal Division with Estimations
- Graph to Grid: How to Completing a Table Using Information from a Graph
- Acids, Bases, and the pH Scale
- Watchdog role
- The Best Grade 8 ELA Practice Tests for Rhode Island Students
- How to Solve Unknown Angles? (+FREE Worksheet!)
- Ratio, Proportion and Percentages Puzzle – Challenge 30
- Full-Length HiSET Math Practice Test-Answers and Explanations
- Free Grade 6 English Worksheets for New York Students
- 49 Free SBAC Grade 3 Math Worksheets for South Dakota Students (One Skill Per Page)
What people say about "Domain and Range of a Relation: Pairs, Graphs and Formulas"?
No one replied yet.