The domain of a function is the set of all possible input values (usually x) for which the function is defined and produces real outputs. Writing the domain involves specifying this set clearly, often using interval notation or set notation. To write the domain of a function:
- Analyze the function to find any restrictions on the input values, such as:
- Denominators that cannot be zero.
- Even roots (like square roots) that require non-negative radicands.
- Logarithms that require positive arguments.
- Express the domain considering these restrictions:
- Use interval notation to describe continuous sets of values (e.g., [0,∞)[0,\infty)[0,∞) for all x≥0x\geq 0x≥0).
- Use set notation to precisely describe exclusions or specific conditions (e.g., {x∈R∣x≠0}\{x\in \mathbb{R}\mid x\neq 0\}{x∈R∣x=0} means all real numbers except 0).
Interval notation conventions:
- Parentheses ()()() denote values not included in the domain (open interval).
- Brackets [][][] denote values included in the domain (closed interval).
- Infinity is always represented with parentheses since it's not a specific number.
For example:
- The domain of f(x)=1xf(x)=\frac{1}{x}f(x)=x1 is all real numbers except 0, written as (−∞,0)∪(0,∞)(-\infty,0)\cup (0,\infty)(−∞,0)∪(0,∞) or {x∈R∣x≠0}\{x\in \mathbb{R}\mid x\neq 0\}{x∈R∣x=0}.
- The domain of g(x)=xg(x)=\sqrt{x}g(x)=x is all real numbers x≥0x\geq 0x≥0, written as [0,∞)[0,\infty)[0,∞).
These two ways (interval and set notation) are both correct and can be used depending on the context or complexity of the domain.
