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Domain Calculator
Last updated: May 31, 2026
Written by Blake Boege
A domain calculator is an algebraic utility used to find the complete set of input values (domain) for which a real-valued mathematical function is defined. It analyzes functions to enforce mathematical constraints, such as ensuring denominators are non-zero, values inside square roots are non-negative, and arguments of logarithms are positive. It outputs the domain in interval notation and lists vertical asymptotes.
Find the allowable domain inputs and vertical asymptotes of common algebraic, rational, and radical functions.
Quick Answer
Determine the domain of a mathematical function. Input f(x) to see allowable real-number inputs in interval notation and vertical asymptotes.
Mathematical Function
Input common functions like 1 / (x - 4), sqrt(x + 3), or ln(x - 5).
e.g. 1 / (x - 4)
Single letter matching the function variable. · e.g. x
Domain of f(x)
(−∞, 4) U (4, ∞)
The domain represents all valid inputs. Denominators cannot equal zero, and square roots must contain values ≥ 0.
Examples
Rational f(x) = 1 ÷ (x − 4)
Domain: (−∞, 4) U (4, ∞) or x ≠ 4
Square root f(x) = √(x + 3)
Domain: [−3, ∞) or x ≥ -3
How it works
The calculator scans the function expression for denominators, square roots, and logarithms, setting up inequality constraints to solve for allowable inputs.
Denominator rule · Denominator ≠ 0
Radical rule · Radical term ≥ 0
Domain rules and restrictions
To find the domain of a real-valued function, look for operations that are undefined:
- Division by Zero: Set the denominator to zero and solve. The roots must be excluded. E.g., for
1/(x−2), solvex−2 = 0 → x = 2. Domain is all x ≠ 2. - Even Roots (Square Roots): The expression inside must be non-negative. Solve
Inside ≥ 0. E.g., for√(x+3), solvex+3 ≥ 0 → x ≥ −3. - Logarithms: The input argument must be strictly positive. Solve
Inside > 0. E.g., forln(x−5), solvex−5 > 0 → x > 5.
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- Inverse Function Calculator — calculate the inverse of mathematical functions.
- Slope Calculator — find the slope and equations of linear segments.
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Frequently asked questions
The domain is the complete set of all possible input values (typically x) for which the function is defined and produces a real, finite output value.
The three most common restrictions are: 1) Divisors cannot be 0. 2) Terms under even roots (like square roots) must be greater than or equal to 0. 3) Logarithm inputs must be strictly greater than 0.
For all polynomial functions (linear, quadratic, cubic, etc.), the domain is all real numbers. It can be written as (−∞, ∞) or R.
Interval notation uses brackets [ ] for values that are included, parentheses ( ) for values that are excluded, and U (union) to combine separate intervals. E.g., x ≠ 3 is written as (−∞, 3) U (3, ∞).
A vertical asymptote is a vertical line x = c where the function's output approaches positive or negative infinity as x approaches c. These values are excluded from the function's domain.
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