Calculator
Example data table
| Case | Domain inequality | Range inequality | Domain interval | Range interval |
|---|---|---|---|---|
| Example 1 | -5 ≤ x < 8 | -2 < y ≤ 10 | [-5, 8) | (-2, 10] |
| Example 2 | x > 3 | y ≤ 12 | (3, ∞) | (-∞, 12] |
| Example 3 | x = 4 | -1 ≤ y ≤ 1 | [4, 4] | [-1, 1] |
Formula used
Interval notation: closed endpoints use brackets, and open endpoints use parentheses. Examples are [a, b], (a, b), [a, b), and (a, b].
Inequality notation: a ≤ x ≤ b, a < x < b, x ≥ a, or x < b describe the same set in inequality form.
Finite interval length: length = upper bound − lower bound.
Finite interval midpoint: midpoint = (lower bound + upper bound) ÷ 2.
Membership check: a test value belongs to the set only when it satisfies both endpoint conditions.
How to use this calculator
- Enter the lower and upper bounds for the domain.
- Select whether each domain endpoint is open or closed.
- Use infinity checkboxes when a bound extends forever.
- Repeat the same setup for the range values.
- Optionally enter x and y test values.
- Choose the decimal precision you need.
- Press Calculate to view interval, inequality, midpoint, and length.
- Use the CSV and PDF buttons to export the current result.
About domain and range inequalities
Why this topic matters
Domain and range inequalities describe where a relation is allowed to exist. The domain lists valid x-values. The range lists valid y-values. These limits appear in algebra, graphing, calculus, and modeling. Clear interval work reduces mistakes during analysis.
Reading interval notation
Brackets mean the endpoint is included. Parentheses mean the endpoint is excluded. A finite interval such as [2, 7) includes 2 but excludes 7. Infinite intervals always use parentheses next to infinity. This rule keeps notation consistent and easy to read.
Connecting inequalities and sets
An inequality and an interval can represent the same set. For example, -3 < x ≤ 5 becomes (-3, 5]. Set-builder notation also expresses the same information. Each form is useful in different tasks. Teachers often switch between them in lessons and exams.
What this calculator returns
This calculator converts your bounds into interval notation, inequality notation, and set-builder notation. It also classifies the result. You can identify a finite interval, semi-infinite interval, single-point interval, empty set, or all real numbers. Test values help confirm membership fast.
Useful checks for students
Always compare the lower and upper bounds before solving. If both finite bounds are equal, open endpoints create an empty set. Closed endpoints create one exact value. For finite intervals, midpoint and length can support graph sketches, estimation, and classroom explanations.
When to use domain and range restrictions
You will use these restrictions when studying piecewise graphs, inverse functions, square roots, rational expressions, and real-world models. Stating valid inputs and outputs improves accuracy. It also helps when writing reports, checking graph windows, and validating mathematical assumptions.
FAQs
1. What is the domain in inequality form?
The domain in inequality form shows which x-values are allowed. It may appear as x ≥ 2, x < 9, or -4 ≤ x < 6. This notation is direct and easy to compare with graph boundaries.
2. What is the range in inequality form?
The range in inequality form shows which y-values can occur. It may be written as y > -1 or 0 ≤ y ≤ 12. It focuses on outputs instead of inputs.
3. What is the difference between open and closed endpoints?
Closed endpoints include the boundary value. Open endpoints exclude it. In interval notation, closed endpoints use brackets. Open endpoints use parentheses. This difference changes membership tests and graph dots.
4. Why does infinity always use parentheses?
Infinity is not a real endpoint that can be reached or included. Because of that, interval notation always places parentheses next to -∞ or ∞. This is a standard rule.
5. Can the calculator show an empty set?
Yes. If the lower and upper finite bounds are equal but one endpoint is open, no number satisfies the condition. The calculator marks that case as an empty set.
6. When is the midpoint available?
The midpoint is available only for non-empty finite intervals. Semi-infinite intervals and all real numbers do not have a finite midpoint. A single-point interval has that same point as its midpoint.
7. What does the test value feature do?
It checks whether a chosen x-value or y-value belongs to the domain or range you entered. This is useful for homework checking, graph interpretation, and quick verification.
8. Can I use this for graphing practice?
Yes. The calculator helps you translate endpoint rules into interval form before plotting. That makes it easier to place open circles, closed circles, and shaded regions correctly.