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Domain

Definition

The domain of a function is its specified set of allowed inputs. We call an input xx the independent variable.

Interpretation

The domain specifies which input values are allowed for a function. It is part of the function itself: the same formula on two different domains gives two different functions.

The upper half of a circle, the graph of the square root of 4 minus x squared, with its shadow on the x-axis from -2 to 2 shaded as the domain; outside it the function is not defined.

The red interval [−2,2][-2,2] is the domain of the function in the figure: it contains the inputs where ff is defined and is the projection of the graph onto the xx-axis. For ∣x∣>2|x|>2 the number 4−x24-x^2 is negative, so its square root is not defined.

When a function is given only by a formula, the usual convention is to take as domain the largest set of real numbers on which the formula makes sense. The usual obstacles are division by zero, even roots of negative numbers and logarithms of numbers that are not positive. The domain also has to be checked when functions are combined: a composition g∘fg\circ f needs the values of ff to lie in the domain of gg, and choosing a smaller domain can make a function invertible, as x2x^2 is from [0,∞)[0,\infty) to [0,∞)[0,\infty).

A common slip is to mix up the domain, which collects inputs, with the range, which collects outputs. For f(x)=4−x2f(x)=\sqrt{4-x^2} the domain is [−2,2][-2,2], but the range is [0,2][0,2].

Key Results

  • A function is only defined on its domain.
  • Restrictions on the domain may arise from algebraic or analytical conditions.
  • Different domains lead to different functions, even if formulas coincide.

Examples

  • The function f(x)=xf(x)=\sqrt{x} has domain [0,+∞).[0,+\infty)\text{.}
  • The function f(x)=1/xf(x)=1/x has domain R∖{0}.\mathbb{R}\setminus\{0\}\text{.}
  • The function f(x)=x2/3=(x3)2f(x)=x^{2/3}=(\sqrt[3]{x})^2 has domain R,\mathbb{R}\text{,} whereas an even root such as x\sqrt{x} requires a nonnegative input.

Prerequisites

Sources

Aliases: domain of definition Symbols: infinity, real numbers, square root

Last updated: 20 October 2026 03:02