### Domain definition

The domain of a function is the set of its possible inputs, i.e., the set of input values where for which the function is defined. In the function machine metaphor, the domain is the set of objects that the machine will accept as inputs.

For example, when we use the function notation $f: \R \to \R$, we mean that $f$ is a function from the real numbers to the real numbers. In other words, the domain of $f$ is the set of real number $\R$ (and its set of possible outputs or codomain is also the set of real numbers $\R$).

If we just define a function such as $f(x)=\sqrt{x-1}$ and don't explicitly state its domain, we typically assume that the domain is the largest subset of real numbers where $f$ could be define. In this example, we'd implicitly understand that the domain is the set of real numbers greater than or equal to 1: $\{ x \,|\, x \ge 1 \}$.