Functions: Domain

The domain of a function is the set of all possible input values (usually $x$-values) for which the function is defined and produces a valid real number.

Domain constraints can arise from physical reality, defined rules, or mathematical operations that are not defined over the real numbers:

• Circle Area (Physical Constraints): The area of a circle as a function of its radius is $A(r) = \pi r^2$. Since a physical radius cannot be negative, the domain is restricted to $r \ge 0$.
• Tax Brackets (Piecewise Constraints): In income tax systems, tax rates are defined using brackets (piecewise intervals). For example: a 10% tax rate for income up to $5,000$ ($0 \le x \le 5000$), and a 15% tax rate for income above $5,000$ ($x > 5000$). Each bracket represents a part of the domain.
• Rational Functions (Division by Zero): For the function $f(x) = \frac{1}{x - 3}$, division by zero is undefined. Since the denominator cannot be $0$, we set $x - 3 \neq 0$, which gives a domain of all real numbers except $x = 3$.
• Radical Functions (Square Roots): For the function $g(x) = \sqrt{x - 2}$, we cannot take the square root of a negative number in real mathematics. The expression under the square root must be non-negative, so $x - 2 \ge 0$, which gives a domain of $x \ge 2$.

A function's domain can be determined by looking at the horizontal extent of its graph. If the domain starts or ends at a specific boundary, the way we draw the endpoint depends on whether the boundary is included:

• Open Circle (small empty circle): Used for strict inequalities ($>$ or $<$) where the boundary point itself is not included. For example, if the domain is $x > 5$, the point at $x = 5$ is drawn as a small open circle.
• Solid Point (normal filled point): Used when the boundary point is included ($\ge$ or $\le$).