Natural domain

Often a function is given through its “formula”, without specifying the domain. In that case the natural domain is understood to be the largest subset of $\mathbb{R}$ on which the formula makes sense.

In brief — The most common constraints for the natural domain

• Denominators: must not vanish.
• Even-index roots: the argument must be $\ge 0$.
• Logarithms (which we shall see shortly): the argument must be $> 0$.
• Absolute value, odd-index roots, polynomials: no constraint (domain $\mathbb{R}$).

The natural domain is the intersection of all the constraints.

Example

Determine the natural domain of $f(x) = \dfrac{\sqrt{x-1}}{x-3}$.

Constraints: $x - 1 \ge 0$ (root) and $x - 3 \ne 0$ (denominator). Intersection: $x \ge 1 \wedge x \ne 3$, that is $\boxed{[1;3)\cup(3;+\infty)}$.

Links

Topics: Functions and properties
Concepts: Domain · Natural domain
Skills: Reasoning by cases · Studying a function