Domain of Function Calculator

Calculator inputs

Function family

Polynomial: f(x) = ax² + bx + c

Rational: f(x) = (ax² + bx + c)/(dx² + ex + f)

Numerator coefficients

Denominator coefficients

Quadratic controller: q(x) = ax² + bx + c

Log base

Square root of rational expression: √((mx + n)/(px + q))

Numerator line

Denominator line

Formula used

Core domain rules

Polynomial functions: domain is all real numbers.
Rational functions: denominator must never equal zero.
Square root functions: radicand must be greater than or equal to zero.
Reciprocal square roots: radicand must be strictly greater than zero.
Logarithmic functions: argument must be positive, with base greater than zero and not equal to one.
Square roots of rational expressions combine inequality testing with denominator restrictions.

Quadratic boundary points

For expressions of the form ax² + bx + c, boundary points come from the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Those points split the number line into intervals where the sign stays consistent.

Function	Key restriction	Domain
f(x) = (x + 2)/(x - 3)	x - 3 ≠ 0	(-∞, 3) ∪ (3, ∞)
f(x) = √(x² - 9)	x² - 9 ≥ 0	(-∞, -3] ∪ [3, ∞)
f(x) = log₁₀(x² - 4)	x² - 4 > 0	(-∞, -2) ∪ (2, ∞)
f(x) = 1/√(x² - 1)	x² - 1 > 0	(-∞, -1) ∪ (1, ∞)
f(x) = x² + 5x + 6	No restriction	(-∞, ∞)

FAQs

1. What is a function domain?

The domain is the set of x-values allowed in a function. It includes every real input that keeps the expression defined and excludes values causing impossible operations, like division by zero or square roots of negative numbers.

2. Why do rational functions exclude some values?

Rational functions contain a denominator. Any x-value making that denominator zero must be removed because division by zero is undefined. Those values become holes or vertical asymptotes, depending on the full expression.

3. Why does a square root need a nonnegative radicand?

Over the real numbers, square roots are defined only when the inside expression is zero or positive. Negative inputs would require complex numbers, so they are excluded from the real-valued domain.

4. Why are logarithm arguments restricted to positive values?

A real logarithm is defined only for positive arguments. Zero and negative values are excluded. The base also matters: it must be positive and cannot equal one.

5. Does a polynomial ever have a restricted domain?

Ordinary polynomials do not have real-number restrictions. Because they use only repeated addition and multiplication, every real x-value works, so the domain is always all real numbers.

6. What do boundary points mean in domain work?

Boundary points are the zeros of denominators, radicands, or logarithm arguments. They split the number line into intervals. Testing one point from each interval tells you whether that whole interval belongs in the domain.

7. Can the graph confirm my interval answer?

Yes. The plotted curve helps you see gaps, missing branches, and restricted regions. It is a visual check, but the interval result should still come from algebraic rules.

8. Can this tool solve every possible domain problem?

This version handles many common classroom patterns well. Extremely advanced expressions may need manual analysis or a computer algebra system, especially when several nested restrictions appear together.