Domain

If $f:X\to Y$ is a function, then the domain of $f$ is the set $X$.

The domain is part of the data of a function. In analysis, changing the domain can change key properties (e.g., injectivity of $x\mapsto x^2$ on $\mathbb{R}$ versus on $[0,\infty)$).

Examples:

• For $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$, the domain is $\mathbb{R}$.
• For $\log:(0,\infty)\to\mathbb{R}$, the domain is $(0,\infty)$.
• The restriction $f|_{[0,1]}$ has domain $[0,1]$ even if $f$ was originally defined on $\mathbb{R}$.