1. What Are Domain and Range?
Every mathematical graph on a Cartesian plane is a collection of coordinate pairs (x, y).
- Domain: The set of all possible input values (all x-coordinates) for which the graph exists. Visually, this is the total horizontal width or span of the curve.
- Range: The set of all resulting output values (all y-coordinates) produced by the function. Visually, this is the total vertical height from bottom to top.
A simple memory trick: D comes before R in the alphabet, just like X comes before Y. Therefore, Domain = X and Range = Y.
2. Decoding Graph Endpoints: Dots, Circles, & Arrows
When inspecting a graph, the endpoints tell you whether boundaries are included or excluded:
Solid Dot (•)
Included. The graph reaches this exact coordinate.
Inequality: ≤ or ≥
Open Circle (○)
Excluded. A hole or open boundary. The graph gets infinitely close but never touches it.
Inequality: < or >
Arrows (∞)
Infinite. The graph continues without bound toward positive or negative infinity.
(-∞, ∞)
3. The 2-Scan Visual Technique
You do not need complicated formulas to find domain and range from a graph. Use this two-step physical scan with your finger or a ruler:
Scan 1: The Left-to-Right Horizontal Scan (For Domain)
Hold a ruler vertically and slide it across the page from far left to far right. Note the very first x-value where your ruler touches the graph (the lower domain bound) and the very last x-value where it leaves the graph (the upper domain bound). If the graph has an arrow pointing left, it starts at -∞. If it has an arrow pointing right, it ends at +∞. Check for any open circles or holes along the way!
Scan 2: The Bottom-to-Top Vertical Scan (For Range)
Hold your ruler horizontally and slide it upwards from the bottom of the grid toward the top. Note the lowest point on the graph (the minimum y-value) and the highest point on the graph (the maximum y-value). Always write the lowest number first! If an arrow points downward, the range begins at -∞; if it points upward, it ends at +∞.
4. Writing Your Answer: Interval vs. Inequality Notation
Teachers and textbooks use different mathematical formats. Here is how the three main notations translate:
| Graph Description | Interval Notation | Inequality Notation | Set-Builder Notation |
|---|---|---|---|
| From -3 (included) to 5 (included) | [-3, 5] | -3 ≤ x ≤ 5 | {x | -3 ≤ x ≤ 5} |
| From -2 (excluded) to 4 (included) | (-2, 4] | -2 < x ≤ 4 | {x | -2 < x ≤ 4} |
| All numbers greater than or equal to 0 | [0, ∞) | x ≥ 0 | {x | x ≥ 0} |
| Continues infinitely in both directions | (-∞, ∞) | -∞ < x < ∞ | {x | x ∈ ℝ} |
5. Step-by-Step Worked Examples
Example 1Line Segment with Solid and Open Endpoints
Suppose a line segment starts at solid dot (-4, -2) and ends at open dot (3, 5).
Domain (X): Leftmost x is -4 (solid → bracket [). Rightmost x is 3 (open → parenthesis )).
→ Domain: [-4, 3) or -4 ≤ x < 3
Range (Y): Lowest y is -2 (solid → bracket [). Highest y is 5 (open → parenthesis )).
→ Range: [-2, 5) or -2 ≤ y < 5
Example 2Parabola Opening Upwards
Consider a standard quadratic curve f(x) = x2 - 3 with vertex at (0, -3) and arrows extending upwards on both sides.
Domain (X): The curve extends left and right forever with arrows.
→ Domain: (-∞, ∞) (All Real Numbers)
Range (Y): The lowest point is the vertex at y = -3 (included → [). The curve goes upward forever (∞).
→ Range: [-3, ∞) or y ≥ -3
Example 3Circle of Radius 5
A circle centered at the origin (0, 0) with a radius of 5 units.
Domain (X): Leftmost edge is x = -5, rightmost edge is x = 5.
→ Domain: [-5, 5] or -5 ≤ x ≤ 5
Range (Y): Lowest edge is y = -5, highest edge is y = 5.
→ Range: [-5, 5] or -5 ≤ y ≤ 5
6. Top 4 Mistakes to Avoid
1. Writing Range from Top to Bottom
Interval notation MUST always be written from lowest to highest: [min, max]. Writing [5, -2] is invalid notation. Always write [-2, 5].
2. Putting a Square Bracket on Infinity
Infinity is an endless concept, not a reachable number. Writing [0, ∞] is incorrect. Always use parentheses on infinity: [0, ∞).
3. Mixing Up X and Y
Accidentally writing domain bounds using y-values, or range bounds using x-values. Remember: Domain is horizontal (X), Range is vertical (Y).
4. Overlooking Internal Holes
If a continuous line has an open circle in the middle at x = 2, you must split your domain into two intervals joined by a union symbol (∪): (-∞, 2) ∪ (2, ∞).
7. Frequently Asked Questions (FAQ)
What is the simplest way to find the domain of a graph?
Scan the graph horizontally from left to right along the X-axis. Find the leftmost x-value where the graph begins and the rightmost x-value where it ends, noting whether endpoints are solid dots (included), open dots (excluded), or arrows (infinity).
What is the simplest way to find the range of a graph?
Scan the graph vertically from bottom to top along the Y-axis. Identify the lowest y-value (minimum) and the highest y-value (maximum). Always write range from the smallest number to the largest number.
When do you use brackets vs parentheses in interval notation?
Use square brackets [ or ] when the endpoint is included (represented by a solid dot or a continuous line). Use parentheses ( or ) when the endpoint is excluded (represented by an open circle, a hole, or infinity / negative infinity).
Can infinity ever have a square bracket in interval notation?
No. Infinity (∞) and negative infinity (-∞) represent boundless directions rather than specific numbers, so they always take parentheses: (-∞, ∞).
How do you find the domain and range of a parabola?
For any vertical parabola f(x) = ax² + bx + c, the domain is all real numbers, (-∞, ∞). The range depends on the vertex: if it opens upward with vertex at (h, k), the range is [k, ∞); if it opens downward, the range is (-∞, k].