Calculator
Example Data Table
| Expression type | Restriction | Interval notation | Reason |
|---|---|---|---|
| Polynomial | None | (-∞, ∞) | Every real input is allowed. |
| Rational | x ≠ 4 | (-∞, 4) ∪ (4, ∞) | The denominator cannot be zero. |
| Square root | x ≥ 5 | [5, ∞) | The radicand cannot be negative. |
| Logarithm | x > -3 | (-3, ∞) | The log argument must be positive. |
Formula Used
The domain rule depends on the expression. For polynomials, the domain is all real numbers. For rational expressions, exclude every value that makes the denominator equal zero. This is written as D = ℝ \ {zeros of denominator}.
For square root expressions, solve the radicand inequality radicand ≥ 0. For logarithmic expressions, solve the argument inequality argument > 0. For custom intervals, use a bracket when an endpoint is included and a parenthesis when it is excluded.
When several restrictions exist, combine them on one real number line. Then write each allowed region from left to right. Join separated regions with the union symbol.
How to Use This Calculator
- Select the calculation mode that matches your expression.
- Enter denominator zeros, listed gaps, or extra exclusions.
- Enter a boundary value for radical or logarithm modes.
- Use custom endpoints when the problem gives an interval directly.
- Press the calculate button and read the result above the form.
- Download the result as CSV or PDF when needed.
Understanding Domain Interval Notation
Domain interval notation gives a compact way to describe allowed input values. It is common in algebra, calculus, graphing, and data modeling. A domain tells where a function is defined. Interval notation tells the same idea with brackets, parentheses, infinity symbols, and union signs.
A bracket means an endpoint is included. A parenthesis means an endpoint is not included. Infinity always uses a parenthesis. The symbol U joins separate allowed intervals. For example, all real numbers except 2 become (-∞, 2) U (2, ∞). That notation shows a gap at x = 2.
This calculator helps with several common cases. Polynomial and linear functions usually accept all real numbers. Rational functions exclude denominator zeros. Square root functions require the radicand to be greater than or equal to zero. Logarithmic functions require the argument to be greater than zero. Custom intervals also work when a graph or problem statement gives endpoints directly.
The tool is useful because domain mistakes are easy to make. A single endpoint can change an answer. A closed endpoint may include a valid solution. An open endpoint may remove it. The calculator shows the interval, set-builder text, restrictions, and a short explanation. This helps users compare answers and spot missing exclusions.
Students can use it before graphing. Teachers can use it to create examples. Analysts can use it when a formula has limits. It is also helpful for checking piecewise domains. Enter the known restriction, choose the rule, and review the output.
The result should still be checked against the original problem. Some functions have combined restrictions. A rational square root may need both a denominator check and a radicand check. A logarithm inside a fraction may need argument and denominator rules. Use the additional exclusions field for these combined cases.
Good domain work improves later steps. It prevents invalid substitutions. It keeps graphs honest. It also supports cleaner interval answers. With practice, interval notation becomes quick and clear.
Many classroom questions hide the domain inside wording. Look for denominators, radicals, logs, graphs, tables, and stated limits. Mark each unsafe value first. Then write the remaining safe values in ordered intervals. This habit reduces errors during tests and homework. It also improves confidence.
FAQs
What is domain interval notation?
It is a compact way to show every input value a function allows. It uses parentheses, brackets, infinity symbols, and union signs to describe the domain.
When should I use brackets?
Use brackets when an endpoint is included in the domain. This usually happens with inequalities like x ≥ 4 or x ≤ 9.
When should I use parentheses?
Use parentheses when an endpoint is not included. Infinity also always uses parentheses because infinity is not a fixed reachable value.
How are rational function domains found?
Set the denominator equal to zero. Any solution is excluded from the domain. The remaining real values form the interval answer.
How are square root domains found?
Set the radicand greater than or equal to zero. Solve that inequality. The solution interval is the real domain for the square root expression.
How are logarithm domains found?
Set the logarithm argument greater than zero. Solve the inequality. The endpoint is open because zero is not allowed inside a logarithm.
Can I enter multiple exclusions?
Yes. Separate values with commas, spaces, or semicolons. The calculator sorts them and creates separate open intervals around each excluded point.
Can combined restrictions be checked?
Yes. Choose the main rule first. Then add extra excluded values. Always compare the result with the original expression for special cases.