Domain and Range Calculator

A domain and range calculator answers two questions: which inputs are allowed, and which outputs actually occur? The first is mechanical once you know the few rules that forbid an input; the second depends on the shape of the graph. This page covers both in the interval notation most courses expect, with steps you can reproduce on paper.

What domain and range mean

Think of a function as a machine. The domain is the set of numbers it accepts without breaking, and the range is the set of numbers it can hand back. Unless a problem states otherwise, the domain is the natural one: every real number for which the formula gives a real result. On a graph, the domain is the curve's shadow on the x-axis and the range its shadow on the y-axis. f(x) = x² − 4x + 1 accepts every real number, so its domain is (−∞, ∞), but it never dips below −3, so its range is [−3, ∞).

The rules that restrict a domain

Only a few operations can reject a real input. Everything else (sums, products, integer powers, odd roots, |x|, eˣ, sin and cos) accepts every real number. Write one condition for each restricted operation, solve it, and keep the x-values that pass all of them:

The inequality signs matter: √0 is fine, so roots use ≥, while ln 0 is undefined, so logarithms use >. When operations combine, every condition applies at once: f(x) = 1/√(4 − x) needs 4 − x ≥ 0 for the root and √(4 − x) ≠ 0 for the denominator, which together leave x < 4.

Worked example: the domain and range of (x + 1)/(x − 2)

Domain. The only restriction is the denominator. Setting x − 2 = 0 gives x = 2, so the domain is every real number except 2:

Range. Write y = (x + 1)/(x − 2) and solve for x. Multiplying out gives yx − 2y = x + 1, so x(y − 1) = 2y + 1 and x = (2y + 1)/(y − 1). That formula produces a valid x for every y except y = 1, where it would divide by zero. So the function never outputs 1, and every other value is reached:

The missing value is the horizontal asymptote, the ratio of the leading coefficients (1 ÷ 1 = 1). A few spot checks agree: f(0) = −1/2, f(3) = 4 and f(5) = 2, and no input lands exactly on 1. The calculator reports the same domain and range and lists the vertical asymptote x = 2 and the horizontal asymptote y = 1 as its reasoning.

Worked example: the domain and range of √(9 − x²)

Domain. The expression under the square root must not be negative:

Both endpoints are included because √0 = 0 is a perfectly good output, so the interval uses brackets.

Range. On its own, 9 − x² takes every value up to its maximum of 9 at x = 0, that is (−∞, 9]. The square root only sees the part from 0 to 9, and it turns [0, 9] into [0, 3]. So the range is [0, 3]: f(0) = 3 is the top, f(−3) = f(3) = 0 is the bottom, and values in between such as f(√5) = 2 fill the gap. The graph is the upper half of the circle x² + y² = 9, which makes both answers easy to see. This is the calculator's default example.

Domain and range of 17 common functions

Every row comes from the calculator above; paste any function from the first column to check the other two.

Three rows deserve a second look. In 1/(x² − 4) the range splits into two pieces because the outputs between −1/4 and 0 never occur. In x/(x² + 1) the horizontal asymptote y = 0 is still part of the range, since f(0) = 0. And (x² − 1)/(x − 1) loses exactly one output, 2, because of a hole at x = 1.

How to find the range of a function

A range of a function calculator has to pick a method that fits the formula, and so do you. Four methods cover most homework:

• Quadratics: use the vertex. For f(x) = ax² + bx + c the vertex sits at x = −b ÷ (2a). For x² − 4x + 1 that is x = 2 with f(2) = −3, so the range is [−3, ∞) because a > 0.
• Rational functions: solve for x. As in the first example, rearrange y = f(x) into x in terms of y and look for forbidden y-values. Then check whether the turning points limit the range further, as they do for x/(x² + 1).
• Transformed parent functions: follow the range. √u always gives [0, ∞), eᵘ gives (0, ∞) and sin u gives [−1, 1]. Multiplying stretches the range and adding shifts it: 3 sin(2x) + 1 runs from 3(−1) + 1 = −2 to 3(1) + 1 = 4.
• Everything else: find the turning points. Set the derivative to 0, evaluate f at those points and at the ends of the domain, and include any values approached near asymptotes. That is what the calculator does numerically when no exact method applies.

Interval notation, set-builder notation and inequalities

An interval notation calculator converts between three ways of writing the same set. A parenthesis means the endpoint is left out, a bracket means it is included, ∪ joins separate pieces, and ∞ always gets a parenthesis because infinity is not a number you can include. Set-builder notation describes the set by a condition, as in {x | x ≥ 3}. The converter tab in the calculator handles all three in both directions; these rows come straight from it:

Trig functions: domains that repeat

Sine and cosine accept every real number, but tan x = sin x ÷ cos x and sec x = 1 ÷ cos x break wherever cos x = 0, at x = π/2 + kπ for every integer k. Those exclusions repeat forever, so the domain is a rule rather than a list: the calculator writes the domain of tan(x) as (−π/2 + kπ, π/2 + kπ) for every integer k, or {x ∈ ℝ | x ≠ π/2 + kπ, k ∈ ℤ}. A coefficient changes the spacing: tan(2x) excludes x = π/4 + kπ/2. Cotangent and cosecant divide by sin x, so they exclude x = kπ. Arcsine and arccosine only accept inputs from −1 to 1, which is why asin(2x − 1) has domain [0, 1] and range [−π/2, π/2].

How to use this calculator

Type the right-hand side of f(x) into the box. Use ^ for powers, sqrt() or √ for square roots, cbrt() for cube roots, abs() or vertical bars for absolute value, and pi for π. log(x) is the common logarithm, ln(x) the natural logarithm and log(x, 2) a base-2 logarithm. You can write 2x or 3(x + 1) without a times sign, but note that 1/2x is read as (1/2)x, following the parser's documented rules; type 1/(2x) for one over 2x. The "Read as" line repeats your function in standard notation so you can catch a misreading before trusting the answer.

The result card shows the domain and range in interval notation with set-builder notation underneath, and marks the range as Exact or Estimate. Below it come a number line, one line per restriction and an explanation of the range. The converter tab turns an inequality such as −2 < x ≤ 5 into interval notation or back. Students use it to check algebra and precalculus homework; teachers use the step list for answer keys.

Key terms

• Domain: all inputs x for which the function gives a real output.
• Range: all outputs y the function actually produces.
• Natural domain: the largest domain the formula allows, used when no domain is stated.
• Vertical asymptote: an excluded x where the function grows without bound, as at x = 2 for (x + 1)/(x − 2).
• Hole: an excluded x where the function approaches a finite value instead, as at (1, 2) for (x² − 1)/(x − 1).
• Interval notation: a set written with parentheses, brackets and ∪, such as (−∞, 2) ∪ (2, ∞).
• Set-builder notation: a set written as a condition, such as {x ∈ ℝ | x ≠ 2}.

Tips for checking your answer

• Find the domain from the original formula, before you simplify anything.
• Test one value just inside and one just outside each endpoint of the domain.
• For the range, try to solve y = f(x) for x; forbidden y-values usually show up immediately.
• Sketch the graph and read both shadows: x-axis for the domain, y-axis for the range.

Limitations: when the range is an estimate

Domains are found with exact rules. The only approximations appear when a boundary has no closed form, such as the single real root of x³ + x + 1, which is shown as ≈−0.6823. Ranges are exact for polynomials, rational functions and single chains of standard functions. For mixed expressions such as x·eˣ or x − ln x, the calculator samples thousands of points, refines every turning point and studies the behavior near asymptotes and at infinity. That works well for smooth functions, but a numeric search cannot prove that a value is never reached, so those ranges carry an Estimate label. A function such as ln(sin(x²)) has a domain made of infinitely many pieces that shrink and never repeat evenly; no list of intervals can hold it, so the calculator says so instead of printing a partial list.

Related calculators

• To solve the inequalities that domain conditions produce, use the Inequality Calculator.
• For the vertex that bounds a quadratic's range, use the Vertex Calculator.
• For the zeros of a denominator or of the expression under a root, use the Quadratic Formula Calculator or the Factoring Calculator.
• To find where a graph crosses the axes, use the X and Y Intercept Calculator.
• For values of logarithms and trig functions, use the Logarithm Calculator and the Trigonometry Calculator.

Sources

Domain and range follow from the definitions of the functions themselves, so every result on this page is computed rather than looked up. The definitions behind the rules:

• NIST Digital Library of Mathematical Functions, §4.2, Definitions: ln x is real for positive x, and log base a is ln z ÷ ln a.
• NIST Digital Library of Mathematical Functions, §4.14, Definitions and Periodicity: tan, csc, sec and cot as quotients of sin and cos, and their periods.
• NIST Digital Library of Mathematical Functions, §4.23, Inverse Trigonometric Functions: the principal values of arcsin, arccos and arctan.
• math.js documentation, Expression syntax: how typed input is parsed, including the implicit multiplication rule behind 1/2x.