Visual Domain and Range — DeltaMath Answers
This module gives you a graph and asks for its domain and range. The reading part is usually fine — people can see where a curve starts and stops. What costs the marks is notation: whether an endpoint takes a bracket or a parenthesis, whether infinity is ever included, and whether a gap in the graph needs a union symbol. This page maps every graph feature to the punctuation it produces.
- Domain is the x-values, read left to right. Range is the y-values, read bottom to top.
- A closed circle includes the endpoint and takes a square bracket. An open circle excludes it and takes a parenthesis.
- An arrow means the graph continues forever — that end becomes
∞or-∞, and infinity always takes a parenthesis. - A visible gap in the graph means two intervals joined with a union symbol
∪.
What the Module Asks
“Visual Domain and Range” shows you a graph — a line segment, a parabola, a piecewise function, sometimes a scatter of discrete points — and asks for the set of inputs it accepts and the set of outputs it produces.
The word visual is doing something specific. Other domain-and-range modules give you an equation and ask you to reason about it algebraically. This one gives you a picture, so every answer is obtained by looking rather than solving. That makes it faster, and it also means the whole difficulty shifts to translating what you see into notation.
Get the reading direction fixed first, because reversing it is a surprisingly common way to lose both parts of a problem at once:
- Domain: sweep left to right. Where does the graph start horizontally, and where does it stop? Those are x-values.
- Range: sweep bottom to top. How low does the graph go, and how high? Those are y-values.
- Always write intervals in increasing order.
[-3, 5], never[5, -3].
Endpoints: Circles and Arrows
Every endpoint on a DeltaMath graph carries a marking, and each marking maps to exactly one piece of punctuation. This table is the module.
| What you see | What it means | Punctuation |
|---|---|---|
| Filled circle ● | The endpoint is part of the graph | Square bracket — [ or ] |
| Open circle ○ | The graph approaches but never reaches it | Parenthesis — ( or ) |
| Arrow → | The graph continues without end | ∞ or -∞, always with a parenthesis |
| A plain stop | Treat as included unless the diagram says otherwise | Square bracket |
| Vertical asymptote | The graph never attains that x-value | Parenthesis, and usually a union |
Infinity never takes a bracket. Not [∞], not (-∞, 5] with a bracket on the left — always (-∞, 5]. Infinity is a direction, not a number the graph reaches, so it can never be “included.” This is close to an automatic mark whenever a graph has an arrow on it.
The Three Notations, Converted
DeltaMath accepts different notations on different problems, and the answer box tells you which one it wants — an interval box has parenthesis and bracket buttons, an inequality box has < and ≤. Same information, three ways of writing it.
| Situation | Interval | Inequality | Set-builder |
|---|---|---|---|
| From -3 to 5, both included | [-3, 5] | -3 ≤ x ≤ 5 | {x | -3 ≤ x ≤ 5} |
| From -3 to 5, neither included | (-3, 5) | -3 < x < 5 | {x | -3 < x < 5} |
| From -3 included, to 5 excluded | [-3, 5) | -3 ≤ x < 5 | {x | -3 ≤ x < 5} |
| Everything from 2 upward, included | [2, ∞) | x ≥ 2 | {x | x ≥ 2} |
| Everything below 4, excluded | (-∞, 4) | x < 4 | {x | x < 4} |
| All real numbers | (-∞, ∞) | x ∈ ℝ | {x | x ∈ ℝ} |
The mapping to memorise is one line long: square bracket goes with ≤ and ≥, parenthesis goes with < and >. Every conversion in the table follows from that.
Gaps, Unions, and Piecewise Graphs
When the graph has a hole in the middle — a piecewise function with a jump, or a rational function with a vertical asymptote — the answer is not one interval. It is two, joined by a union.
The union symbol ∪ means “and also.” A graph running from -5 to 2, missing 2 entirely, then resuming from 2 to 8, has domain [-5, 2) ∪ (2, 8]. Both parentheses around the 2 say the same thing from either side: this value is not in the domain.
A hole in a curve
A single open circle mid-graph. Split the interval at that x-value with parentheses on both sides of it.
A jump between pieces
Two separate segments at different heights. The domain may still be continuous even when the range needs a union — check each separately.
A vertical asymptote
The graph shoots off near a value it never attains. That value is excluded from the domain with parentheses either side.
The trap here is answering with a single interval because the graph looks continuous at a glance. Scan for open circles deliberately before deciding it is one piece — they are small, and on a jump the two pieces often nearly line up.
Why Range Is Harder Than Domain
Domain is a left-to-right sweep and rarely surprises anyone. Range catches people because the highest and lowest points of a graph are not always at its ends.
On a parabola opening upward, the graph extends infinitely far up on both sides, but there is a floor — the vertex. The range starts at that vertex y-value, not at the y-value of either visible endpoint. A parabola with vertex at (1, -4) opening upward has range [-4, ∞) regardless of how wide the drawing is.
- Find the turning point before answering range. On any curve, check the vertex or peak, not just the endpoints.
- Watch for a graph that doubles back. If the curve rises then falls, its maximum is in the middle.
- Horizontal asymptotes bound the range. A curve flattening toward
y = 3without touching it excludes 3. - Discrete points have discrete range. A scatter of dots gives a list, written
{1, 4, 7}, not an interval.
Typing It So the Box Accepts It
DeltaMath renders these answers through a maths input, and a few entry details cause rejections that have nothing to do with your reading of the graph.
- Use the on-screen buttons for
∞and∪. Typing the word “infinity” or the letter U will not be recognised. - Negative infinity needs the minus sign, entered as
-∞rather than typed as text. - Do not add spaces inside brackets.
[-3,5]and[-3, 5]both usually pass, but stray characters do not. - Check which variable it wants. Inequality answers for domain use x; for range they use y. Submitting a range in terms of x is marked wrong even with correct numbers.
Five Graphs, Read
| The graph | Domain | Range |
|---|---|---|
Segment from a filled dot at (-2, 1) to a filled dot at (6, 5) | [-2, 6] | [1, 5] |
Segment from an open circle at (0, 0) to a filled dot at (4, 8) | (0, 4] | (0, 8] |
Upward parabola, vertex (1, -4), arrows on both arms | (-∞, ∞) | [-4, ∞) |
Line with an arrow on the right only, starting at a filled dot (3, 2) | [3, ∞) | [2, ∞) |
Two pieces: [-5, 2) then (2, 8], both rising | [-5, 2) ∪ (2, 8] | Read each piece separately, then union |
Row three is the one worth sitting with. The domain is every real number because the arms extend forever sideways, while the range stops dead at the vertex. Domain and range on the same graph can have completely different shapes, and treating them as a matched pair is a reliable way to get one of them wrong.
How Delta Genie Reads a Graph
Graph-reading is where a page-based tool has a real structural advantage over a photo app or a chatbot. Delta Genie works from the rendered graph in your DeltaMath assignment, so it sees the endpoints, the circle fills, the arrows and the asymptotes directly — it is not trying to reconstruct a picture from a description you typed out.
From there it applies the mapping in this page: fill state to bracket or parenthesis, arrows to infinity, gaps to unions, and then enters the result using the input's own symbol buttons rather than typing characters the box will reject.
Accuracy and timing controls apply here as everywhere. Graph-reading problems go quickly by hand once you know the notation, so a realistic pace on this module is faster than on proofs — but instant and flawless is still instant and flawless.
Domain and Range — FAQ
What is the difference between an open and a closed circle?
A closed or filled circle means the endpoint is part of the graph, so it takes a square bracket in interval notation. An open circle means the graph approaches that point without reaching it, so it takes a parenthesis.
Does infinity get a bracket or a parenthesis?
Always a parenthesis. Infinity is a direction rather than a value the graph attains, so it can never be included as an endpoint. Write (-∞, 5] or [2, ∞), never with a bracket against the infinity side.
How do I write domain and range when the graph has a gap?
As two intervals joined by the union symbol ∪. A graph running from -5 up to but not including 2, then resuming after 2 to 8, has domain [-5, 2) ∪ (2, 8], with parentheses on both sides of the excluded value.
Why is my range wrong when my domain is right?
Usually because the highest or lowest point of the graph is not at its ends. On a parabola the vertex sets the boundary of the range even though the arms extend infinitely. Find the turning point before answering.
Should I answer in interval or inequality notation?
Whichever the answer box is set up for. An interval box shows bracket and parenthesis buttons; an inequality box shows less-than and greater-than symbols. Both describe the same set, but DeltaMath grades the format it asked for.
Is there an answer key for visual domain and range?
No. The graph is regenerated per student with different endpoints and fills, so any shared answer describes a different picture. The endpoint-to-punctuation table on this page is what carries across versions.