Build an Interval
Choose a method, enter your condition, and calculate a formatted interval.
Example Data Table
| Input | Meaning | Interval notation |
|---|---|---|
x ≥ 3 |
Three is included, then values continue right. | [3, ∞) |
-2 < x ≤ 5 |
Negative two is excluded. Five is included. | (-2, 5] |
x < 1 or x ≥ 4 |
Two separate groups of allowed values exist. | (−∞, 1) ∪ [4, ∞) |
x != 0 |
Every real value except zero is allowed. | (−∞, 0) ∪ (0, ∞) |
sqrt(x - 9) |
The square-root input cannot be negative. | [9, ∞) |
Formula Used
Interval notation turns boundary statements into brackets, parentheses, and unions. The relation signs determine endpoint symbols.
For a square root, keep the radicand at least zero. For a logarithm, keep its argument greater than zero. For a denominator, remove values that make it zero.
How to Use This Calculator
- Select an inequality, a supported domain helper, or manual bounds.
- Choose the variable shown in your work.
- Enter the condition or function expression.
- Use manual boundaries when an expression needs custom endpoint choices.
- Add removed points only when they are not already excluded.
- Choose the displayed decimal precision.
- Press the conversion button and review the interval above the form.
- Use CSV or PDF after a result appears.
Reading Functions Through Intervals
Why interval notation matters
Interval notation gives a compact way to describe allowed values. It often appears when finding domains, ranges, solution sets, and graph restrictions. A function can be defined for every real input, or only part of the number line. The notation makes those limits visible. It also prevents long inequality statements from becoming repetitive.
Each interval has two boundary decisions. First, decide where the values begin. Next, decide where they end. A bracket includes its nearby value. A parenthesis excludes it. Infinity always uses a parenthesis. Infinity is not a reachable number, so it cannot be included.
Start with the condition
Read the relation sign before writing symbols. The condition x ≥ 4 includes four. Its interval is [4, ∞). The condition x > 4 excludes four. Its interval is (4, ∞). A two-sided condition needs both decisions. For example, -1 < x ≤ 6 becomes (-1, 6].
Words can reveal the same structure. “At least” means included. “More than” means excluded. “No greater than” means included on the upper side. “Less than” means excluded on the upper side. Translate the wording into an inequality first. Then transfer the inequality into interval notation.
Combine separate regions
Some functions allow two or more disconnected regions. Use the union symbol when either region works. For x < -2 or x ≥ 3, the result is (−∞, -2) ∪ [3, ∞). The union says that a value may belong to the first interval or the second interval.
Exclusions also create separated regions. A rational function may be valid everywhere except one denominator zero. If x = 5 makes a denominator zero, its domain becomes (−∞, 5) ∪ (5, ∞). Both parentheses are needed because five is not allowed. An extra excluded point has the same effect.
Use function rules carefully
Function domains often come from operation rules. Square roots require nonnegative radicands. Logs require positive arguments. Fractions require nonzero denominators. Apply every rule together. The final domain is their intersection. For example, sqrt(x - 1) / (x - 4) needs x ≥ 1 and x ≠ 4. The final interval is [1, 4) ∪ (4, ∞).
Range questions use the same notation, but the values describe outputs. Determine the lowest or highest output first. Check whether the function reaches that value. Then select brackets or parentheses. A graph can help confirm whether a boundary point is filled, open, or never reached.
Check before you submit
Test a value inside every listed interval. It should satisfy the original condition. Test each endpoint too. Included endpoints must work. Excluded endpoints must fail. Finally, test a value in every gap. It should fail. These quick checks catch most notation mistakes.
Use the calculator as a formatting and review tool. For advanced expressions, simplify the math first. Then enter a clear condition or use manual bounds. Keep your original work beside the result during review. That comparison reveals whether a sign, boundary, or domain restriction changed in the translation process itself. This keeps the final interval accurate and easy to read.
Frequently Asked Questions
1. When do I use brackets?
Use brackets when an endpoint belongs to the solution. Inequalities with ≤ or ≥ include their boundary. For example, x ≥ 2 becomes [2, ∞).
2. When do I use parentheses?
Use parentheses when an endpoint does not belong to the solution. Inequalities with < or > exclude their boundary. Infinity always uses a parenthesis.
3. Can an interval include infinity?
No. Infinity represents unbounded direction, not an actual endpoint value. Write parentheses beside −∞ and ∞ in every interval.
4. What does the union symbol mean?
The union symbol, ∪, joins separate solution regions. It means a value can come from one interval or another interval.
5. How is x != 4 written?
Write (−∞, 4) ∪ (4, ∞). The two pieces show that every real value works except four.
6. Does the calculator solve every function domain?
No. The domain helper covers common linear roots, logarithms, and parenthesized linear denominators. For advanced algebra, simplify the function and enter the resulting condition manually.
7. How do square roots affect a domain?
The expression inside a real square root must be zero or greater. Solve that condition, then write the matching interval.
8. How do logarithms affect a domain?
A logarithm argument must be greater than zero. Solve the strict inequality, then use a parenthesis at the boundary.
9. Why might a rational function need a union?
A denominator zero creates a forbidden point. The domain splits around that point, so two intervals joined by a union are needed.
10. Can I add extra excluded values?
Yes. Enter comma-separated numeric points. The calculator removes each listed point from the calculated interval after it builds the main solution.
11. How can I check my final interval?
Test interior values, endpoints, and gaps against the original condition. Clear intervals make domain decisions easier for students daily.