This domain and range calculator finds the domain and range of the standard function families — quadratics, square roots, rationals, absolute values and reciprocal squares — in vertex/shift form, with the reasoning spelled out.
Domain Asks “What Can Go In?” — Range Asks “What Can Come Out?”
The quadratic engine this leans on: our Quadratic Formula Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- f(x) = (x − 2)² − 4 (defaults): vertex (2, −4), opens up → domain: all reals · range: y ≥ −4
- f(x) = √(x − 3) (sqrt, a 1, h 3, k 0): domain x ≥ 3 · range y ≥ 0 — the radicand writes the domain
- f(x) = 1/(x − 2) + 2 (rational, h 2, k 2): domain x ≠ 2 · range y ≠ 2 — both exclusions are the shifts themselves
Finding Any Domain by Hand
- Denominators: set them ≠ 0 and exclude those x values.
- Even roots (√, ∜): the radicand must be ≥ 0 — solve the inequality.
- Logarithms: the argument must be > 0.
- Range from domain + shape: find vertices, asymptotes and endpoints; the range is what the graph actually attains — not what it approaches.
Related Tools
- Quadratic Formula Calculator — roots of the quadratic
- Scientific Notation Calculator — rewrite very large & small outputs
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the domain and range of a function?
Domain: start with all real numbers, then exclude inputs that break the function — denominators equal to zero, negative radicands under even roots, non-positive logarithm arguments. Range: determine the outputs the function attains, usually via its vertex, asymptotes or endpoints. This calculator does both for the standard families in shift form.
What is the domain and range of a quadratic?
Domain: all real numbers, always. Range: set by the vertex — for a(x − h)² + k it is y ≥ k when a > 0 (opens up) and y ≤ k when a < 0. Example: (x − 2)² − 4 has range y ≥ −4.
What is the domain of a square root function?
Wherever the radicand is non-negative: √(x − h) + k has domain x ≥ h and range y ≥ k (for positive a). Negative inputs don’t produce real outputs — that single constraint generates the whole answer.
Why is a value excluded from a range but still “approached”?
Asymptotes: 1/(x − 2) + 2 outputs get arbitrarily close to 2 but the fraction would need to equal 0 to reach it — impossible with a finite denominator. Approached-but-never-attained values are excluded from the range, written y ≠ 2 here.
What does interval notation mean?
Parentheses exclude an endpoint, brackets include it: [3, ∞) means x ≥ 3; (−∞, 2) ∪ (2, ∞) means all reals except 2. Infinity always takes a parenthesis — it is a direction, not a value that can be included.
Is this domain and range calculator free?
Yes — free, instant, no sign-up.