Domain and Range

A rigorous guide to domains and ranges from formulas, graphs, compositions, inverses, and contexts.

A function connects an allowed input set to a resulting output set. A curve appears with a highlighted horizontal domain span and vertical range span. The two labelled spans distinguish allowed inputs from resulting outputs.

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.

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.