Domain of a Function
Functions & Advanced AlgebraThe domain of a function is the complete set of all possible input values for which the function is defined.
Definition
The domain is the set of all allowed inputs for a function. It is every $x$-value you are permitted to plug in.
Example
For the function "square root of $x$," you can only plug in $0$ or positive numbers. Negative numbers are not allowed, so the domain is all numbers greater than or equal to $0$.
Key Insight
Think of the domain as the guest list for a party: only those on the list (allowed inputs) can get in and receive an output.
Definition
The domain of a function is the set of all $x$-values for which the function produces a real, defined output. Common restrictions: denominators cannot be zero, and expressions under even radicals must be non-negative.
Example
For $f(x) = 1/(x - 3)$, $x$ cannot equal $3$ (division by zero). Domain: all real numbers except $3$, written $(-\infty, 3) \cup (3, \infty)$. For $g(x) = \sqrt{x + 2}$, $x + 2 \ge 0$, so $x \ge -2$. Domain: $[-2, \infty)$.
Key Insight
When no domain is stated, assume the natural domain: the largest set of real numbers for which the expression is defined.
Definition
For a function $f: A \to B$, the domain is the set $A$. In analysis, the natural domain of a real-valued function is the maximal subset of $\mathbb{R}$ on which the expression is well-defined. For complex functions, domains extend to subsets of $\mathbb{C}$, and analytic continuation can extend domains further.
Example
The gamma function extends the factorial to all complex numbers except non-positive integers, demonstrating how the domain concept applies beyond elementary functions.
Key Insight
In topology, the domain carries a topological structure. Continuity, differentiability, and integrability are all properties stated relative to the domain.