Domain and Range

Precise definition

The domain of a function is its permitted input set; the range is the set of outputs actually attained. Both are part of the function, not decorations. A formula's natural real domain excludes undefined operations such as zero denominators, even-root negatives, and nonpositive logarithm arguments.

Notation and mathematical language

Use interval notation with round endpoints for exclusion and square endpoints for inclusion, and set-builder notation for complex restrictions. For a composition $f\circ g$, inputs must lie in the domain of $g$ and produce $g(x)$ in the domain of $f$.

Conceptual picture

Domain asks where a machine can accept input; range asks which outputs it can produce. Transformations move restrictions and extrema. Solving $y=f(x)$ for $x$ can reveal range restrictions, but branch and domain conditions must be retained.

Conditions and key results

Context can narrow the mathematical domain, such as nonnegative time or whole item counts. A graphing window is not the domain. A simplified expression may conceal a hole inherited from the original function.

A reliable strategy

1. Start with stated context and scan the original formula for denominators, radicals, logs, inverse trig, and composition restrictions.
2. Solve all input inequalities or exclusions and intersect them.
3. Find range through graph features, transformations, extrema, monotonicity, or solving $y=f(x)$.
4. Verify endpoints, holes, asymptotes, and whether values are attained.

Fully worked example

Interpretation and application

Domain and range encode feasible inputs and attainable outcomes in modelling. An algebraic domain may include negative time, but a physical model may not. State whether restrictions are mathematical, contextual, or both.

Common mistakes

Verification and reasonableness

• Substitute boundary and nearby values.
• Solve $y=f(x)$ and check which $y$ permit an allowed $x$.
• Compare algebraic results with a graph that shows holes and asymptotes explicitly.

Practice

1. Find domain of $1/(x+3)$.
2. Find domain of $\sqrt{x-4}$.
3. Find range of $(x-2)^2+1$.

Further deduction

For $h(x)=\sqrt{x-1}/(x-3)$, restrictions combine rather than compete: $x\ge1$ from the radical and $x\ne3$ from the denominator. Thus the domain is $[1,3)\cup(3,\infty)$. Listing only the most visible restriction is a common source of incomplete domains.

Composition makes domain restrictions operational. For $f(x)=\sqrt{x}$ and $g(x)=x-3$, $(f\circ g)(x)=\sqrt{x-3}$ has domain $x\ge3$ because the output of $g$ must lie in the domain of $f$. In the reverse order, $(g\circ f)(x)=\sqrt{x}-3$ has domain $x\ge0$; composition is not commutative. An inverse relation exchanges input and output coordinates, so domain and range swap. A function has an inverse function on a domain only when it is one-to-one there. Thus $x^2$ on all real numbers fails the horizontal-line test, while its restriction to $x\ge0$ has inverse $\sqrt{x}$. State the restriction with the inverse: omitting it makes $\sqrt{x^2}=x$ falsely universal, since the correct real identity is $\sqrt{x^2}=|x|$.

For a rational inequality, critical numbers include both numerator zeros and denominator exclusions. A sign chart tests intervals between them; numerator zeros may be included for non-strict inequalities, while denominator zeros are never included regardless of the inequality symbol.

Related topics

• Graphs of Functions
• Notation Reference
• Function Composition