Domain of a function calculator, with the conditions

Where a function can be computed, and where it cannot

The domain of a function is the set of real numbers for which the expression gives a real number. It is not a property to be guessed but a list of conditions to be written down: each part of the expression that can fail sets one, and the domain is where they all hold at once — a system, not a single inequality.

Four parts can fail. A denominator must not be zero. The argument of a square root, or of any even root, must not be negative. The argument of a logarithm must be strictly positive: ln 0 does not exist, and neither does the logarithm of a negative number. The argument of arcsine and arccosine must lie between −1 and 1. Odd roots, exponentials, sine, cosine and polynomials set no condition at all.

The brackets carry the difference between ≥ and >. The radicand of √(x − 2) may be zero, so 2 belongs to the domain and the bracket is square: [2, +∞). The argument of ln(x − 2) may not, so 2 is excluded and the bracket is round: (2, +∞). Infinity always takes a round bracket, because it is not a number the function can be evaluated at.

Common mistakes

Writing the condition for a root in the denominator as ≥ 0: in 1/√x the radicand also must not be zero, so the condition is x > 0.
Simplifying before finding the domain. (x² − 4)/(x − 2) simplifies to x + 2, but the original function is not defined at x = 2, and that point stays out of the domain.
Putting a condition on an odd root. ∛x is defined for every real x, negative numbers included: ∛(−8) = −2.
Solving the conditions separately and taking the union instead of the intersection: x must satisfy all of them at once.

Frequently asked questions

How do you find the domain of a function?

Write one condition for each part of the expression that can fail — denominators ≠ 0, radicands of even roots ≥ 0, arguments of logarithms > 0, arguments of arcsine and arccosine between −1 and 1 — and solve them as a system. The domain is the set of x that satisfies them all.

What is the domain of a polynomial?

The whole real line, ℝ: a polynomial is built from sums and products only, and those can always be computed. The same holds for exponentials, sine, cosine and odd roots.

When is a bracket round and when square?

Square when the end belongs to the domain, round when it does not. The zero of a radicand is included, the zero of a denominator or of a logarithm's argument is not, and ±∞ always takes a round bracket.

Why is the domain of tan x not the whole line?

Because tan x = sin x / cos x, and cos x is zero at π/2 and at every odd multiple of it. The domain is every real number except x = π/2 + kπ, with k an integer.

How this calculation works

Conditions of existence: for a/b, b ≠ 0; for an even root ⁿ√a, a ≥ 0 (a > 0 if the root is in a denominator); for ln a and logarithms in any base, a > 0; for arcsin a and arccos a, −1 ≤ a ≤ 1; for tan a, cos a ≠ 0; for a power with a variable exponent, a positive base. The conditions are solved numerically on a fine grid, and each boundary is included or excluded according to its condition: a radicand may reach zero and an arcsine's argument may reach ±1, a denominator and a logarithm's argument may not. Each boundary is then written in exact form when it is a fraction, a root, a multiple of π or a root of a quadratic.

Study of a function

Domain, sign, limits, asymptotes, turning points, concavity and graph, in one go.

Limits

One-sided and two-sided limits, at a point or at infinity, with the indeterminate form.

Asymptotes

Vertical, horizontal and oblique asymptotes, with the limits that give them.

Derivatives

The derivative of a function and the rules used to get it.