Function Domain Calculator

What Is the Domain of a Function?

The domain of a function is the set of all input values x for which the function gives a real output. For f(x) = 1/(x − 3), every real number works except x = 3, where the denominator is 0, so the domain is (−∞, 3) ∪ (3, ∞).

Unless a problem says otherwise, the domain is the largest set of real numbers for which the formula makes sense, often called the natural or implied domain.

How to Find the Domain

Start with all real numbers. Then look for each operation that can fail and write its condition: denominators ≠ 0, expressions under an even root ≥ 0, logarithm arguments > 0. Solve each condition, then keep only the x-values that satisfy all of them at once.

For f(x) = √(x − 2)/(x − 5): the root needs x − 2 ≥ 0, so x ≥ 2, and the denominator needs x − 5 ≠ 0, so x ≠ 5. Together: [2, 5) ∪ (5, ∞).

Domain of Polynomial Functions

Polynomials such as x² + 3x − 2 or 3x − 7 involve only addition, subtraction and multiplication, which work for every real number. Their domain is always all real numbers, (−∞, ∞). The same is true for eˣ, sin(x), cos(x), arctan(x), |x| and odd roots such as ∛x.

Domain of Rational Functions

A rational function is a fraction of polynomials. Set the denominator equal to 0, solve, and exclude those values. For 1/(x² − 9), x² − 9 = (x − 3)(x + 3) = 0 at x = 3 and x = −3, so the domain is (−∞, −3) ∪ (−3, 3) ∪ (3, ∞).

Do not simplify first. (x² − 4)/(x − 2) reduces to x + 2, but the original function divides by zero at x = 2, so x = 2 is still excluded. The graph of the original function has a hole there.

Domain of Radical Functions

For a square root, or any even root, the expression inside must be greater than or equal to 0. √(x − 4) needs x − 4 ≥ 0, so the domain is [4, ∞); √(5 − x) needs x ≤ 5, so (−∞, 5].

When the inside is a quadratic or a fraction, solve the inequality with a sign chart. √(x² − 4) needs x ≤ −2 or x ≥ 2. If the root is in a denominator, as in 1/√(x − 1), the inside must be strictly positive, so the domain is (1, ∞). Odd roots have no restriction.

Domain of Logarithmic Functions

ln(x) and log(x) are defined only for positive inputs, so the argument must be strictly greater than 0. ln(x + 3) needs x > −3, giving (−3, ∞). ln(x² − 4) needs x² − 4 > 0, giving (−∞, −2) ∪ (2, ∞).

Unlike a square root, the argument cannot equal 0, so logarithm boundaries are always open (parentheses, not brackets).

Domain of Trigonometric Functions

sin(x) and cos(x) are defined for every real number. tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x) are undefined where cos(x) = 0, that is x = π/2 + kπ for any integer k. cot(x) = cos(x)/sin(x) and csc(x) = 1/sin(x) are undefined where sin(x) = 0, x = kπ.

The inverse functions arcsin(x) and arccos(x) only accept inputs from −1 to 1, while arctan(x) accepts every real number.

Combining Domain Restrictions

When a function has several restrictions, all of them must hold at the same time, so intersect the solution sets. For ln(x − 2)/(x − 5): the logarithm needs x > 2 and the denominator needs x ≠ 5, so the domain is (2, 5) ∪ (5, ∞).

Work from the inside out with nested functions. For ln(√(x + 1)), the root needs x + 1 ≥ 0 and the logarithm needs √(x + 1) > 0, which means x + 1 > 0. Together the domain is (−1, ∞).

Domain in Interval Notation

Interval notation uses square brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints and for ±∞. [2, 5) means 2 ≤ x < 5. The symbol ∪ joins separate pieces, so (−∞, 3) ∪ (3, ∞) means every real number except 3. An empty domain is written ∅.

Domain in Set-Builder Notation

Set-builder notation describes the domain by its condition: {x ∈ ℝ | x ≥ 2, x ≠ 5} reads “all real x such that x is at least 2 and not equal to 5.” It is especially convenient when a domain is all real numbers except a few values, or has infinitely many exclusions, such as {x ∈ ℝ | x ≠ π/2 + kπ, k ∈ ℤ} for tan(x).

Worked Domain Examples

x² + 2x + 1: a polynomial, so the domain is (−∞, ∞).

1/(x − 3): x − 3 ≠ 0, so x ≠ 3 and the domain is (−∞, 3) ∪ (3, ∞).

√(x − 2): x − 2 ≥ 0, so the domain is [2, ∞).

ln(x + 4): x + 4 > 0, so the domain is (−4, ∞).

√(x + 1)/(x − 2): x ≥ −1 and x ≠ 2, so the domain is [−1, 2) ∪ (2, ∞).

√((x + 1)/(x − 2)): the fraction must be ≥ 0 and x ≠ 2. A sign chart gives x ≤ −1 or x > 2, so the domain is (−∞, −1] ∪ (2, ∞).

Common Domain Mistakes

Cancelling a factor before finding the domain, which loses excluded values such as x = 2 in (x² − 4)/(x − 2). Using ≥ 0 for a logarithm (it must be > 0). Forgetting that a square root in a denominator must be strictly positive. Solving √ and denominator restrictions separately but then listing them instead of intersecting them. Writing a bracket next to ∞, or a bracket at a value that is excluded. Confusing the domain (allowed x) with the range (resulting y).