1. Defining Discrete vs. Continuous Graphs
Before writing domain and range, you must first recognize the fundamental structure of the data:
Isolated Points (Scatter Plots)
Consists of distinct, separate dots with empty space between them. The points are not connected because the intermediate values do not exist in reality.
Connected Lines & Smooth Curves
Consists of unbroken segments or curves where all intermediate decimal and fractional values exist without interruption.
2. How to Write Domain & Range for Discrete Graphs
When stating the domain and range of a discrete scatter plot or set of points:
Rule 1: Always Use Curly Set Braces { }
Never use parentheses ( ) or square brackets [ ]. Enclose all values within curly brackets: { ... }.
Rule 2: List Values in Ascending Order
Standard mathematical convention requires listing numbers from smallest to largest: {-5, -1, 2, 8}.
Rule 3: Eliminate Duplicates
If multiple coordinates share the same x-value or y-value, write that number only once in the set.
3. Side-by-Side Worked Examples
Discrete ExampleScatter Plot with 5 Distinct Points
Given the points plotted on a grid: (-3, 4), (-1, 2), (0, 4), (2, -1), (5, 6).
1. Extract X-values: -3, -1, 0, 2, 5
→ Domain: {-3, -1, 0, 2, 5}
2. Extract Y-values: 4, 2, 4, -1, 6 (Note that 4 appears twice!)
3. Remove duplicate 4 and sort: -1, 2, 4, 6
→ Range: {-1, 2, 4, 6}
Continuous ExampleConnected Segment Between the Same Outer Bounds
A solid line drawn connecting from (-3, -1) to (5, 6) without any breaks.
Every single number between -3 and 5 exists on the line.
→ Domain: [-3, 5] (or -3 ≤ x ≤ 5)
Every single number between -1 and 6 exists on the line.
→ Range: [-1, 6] (or -1 ≤ y ≤ 6)
4. Real-World Context: Why This Distinction Matters
In practical data visualization and science, choosing between discrete and continuous graphs dictates the validity of your conclusions:
| Scenario | Graph Type | Why It Is Classified This Way |
|---|---|---|
| Number of pizzas bought vs. total cost | Discrete | Pizzerias do not sell 3.42 pizzas. Domain is {1, 2, 3, ...}. |
| Distance traveled by a marathon runner vs. time | Continuous | Time and distance flow continuously; you can measure at 12.356 minutes. |
| Number of workers hired vs. widgets produced | Discrete | Workers are whole individuals; you cannot hire 4.7 workers. |
5. Frequently Asked Questions (FAQ)
How can you tell if a graph is discrete or continuous?
A discrete graph consists of separate, disconnected points or dots with spaces in between. A continuous graph consists of an unbroken, connected line or curve that you can trace without lifting your pencil.
Why can't you use interval notation [a, b] for a discrete graph?
Interval notation [a, b] includes every single real number between a and b, including infinite fractions and decimals (like 2.1, 2.01, 2.001). Since discrete graphs only contain specific distinct values, you must list them individually inside set braces { }.
Do you repeat numbers in discrete domain and range sets?
No. In set notation, every unique value is written only once. If two distinct points have the same y-value (such as (2, 5) and (4, 5)), 5 is written only once in the range set.
In what order should discrete domain and range values be written?
Values in set notation should be written in ascending numerical order, from least to greatest: {-4, -1, 0, 3, 7}.
What is an example of a real-world discrete graph?
A graph showing the number of concert tickets sold versus total ticket revenue. You can only buy whole integer tickets (1, 2, 3), not 2.75 tickets.