Explore domain rules across common function families. Enter coefficients, inspect interval notation, and compare restrictions. Visualize valid inputs on clear graphs for faster checks.
Choose a function family, enter coefficients, and press calculate. The domain result appears above this form after submission.
The graph is drawn only on valid x-values. Undefined parts remain broken or hidden.
| Example | Function | Domain Rule | Domain |
|---|---|---|---|
| Polynomial | f(x) = x³ - 2x² + 3x - 4 | No restriction applies. | (-∞, ∞) |
| Rational | f(x) = (x² - 1)/(x² - 9) | Exclude denominator zeros x = ±3. | (-∞, -3) ∪ (-3, 3) ∪ (3, ∞) |
| Even root | f(x) = √(x² - 5x + 6) | Require x² - 5x + 6 ≥ 0. | (-∞, 2] ∪ [3, ∞) |
| Logarithmic | f(x) = log₁₀(x² - x - 6) | Require x² - x - 6 > 0. | (-∞, -2) ∪ (3, ∞) |
| Composite | f(x) = √(x² - x - 6)/(x - 2) | Require radicand ≥ 0 and x ≠ 2. | (-∞, -2] ∪ (3, ∞) |
For a polynomial, the domain is all real numbers because no denominator, logarithm, or even root creates a restriction.
Find the denominator and solve D(x) = 0. Every real solution of that equation must be removed from the domain.
For ⁿ√R(x) with even n, solve R(x) ≥ 0. The domain contains every x-value that keeps the radicand nonnegative.
For logb(A(x)), the base must satisfy b > 0 and b ≠ 1. Then solve A(x) > 0.
When several restrictions appear together, the domain is the intersection of all valid sets. Keep only x-values that satisfy every condition.
The domain is the set of all input values that make a function valid. It excludes values causing division by zero, invalid roots, or invalid logarithm arguments.
A rational function becomes undefined when its denominator equals zero. Those x-values are removed, even if the numerator is also zero there.
If the radicand equals zero, the even root is still real. That is why radical domains use ≥ 0, not just > 0.
A logarithm requires a strictly positive argument. Zero is not allowed, so the calculator solves argument > 0 instead of argument ≥ 0.
The numerator affects graph shape and zeros, but it usually does not remove domain values. The denominator creates the restriction.
That means no real input satisfies the required conditions. The calculator reports the empty set symbol ∅ for that case.
Breaks appear where the function is undefined or outside the domain. This helps visualize exclusions, open intervals, and vertical discontinuities.
Yes. The output is formatted as interval notation and can be copied into notes, worksheets, or study material after review.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.