Math101 · Independent Ontario learning library
Math101Review of Functions
A rigorous, example-driven guide to review of functions, including hypotheses, method choice, verification, and practice.
The central idea
A function assigns each input in its domain exactly one output. Important data include domain, range, intercepts, zeros, symmetry, transformations, composition, and inverse behavior. The composition $(f\circ g)(x)=f(g(x))$ requires $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.
Definitions, hypotheses, and notation
Domain restrictions survive algebraic simplification because a function includes both a rule and its permitted inputs. The expressions $(x^2-1)/(x-1)$ and $x+1$ agree where the first is defined, but they are not identical functions on the real numbers. This distinction later separates a removable discontinuity from an ordinary point on a line.
Inverse functions interchange inputs and outputs, so their graphs reflect across $y=x$. A horizontal-line test identifies whether an inverse can be a function. Restricting $x^2$ to $x\ge0$ yields the inverse $\sqrt x$; without that restriction, the symbol $\pm\sqrt x$ would assign two outputs and fail the function definition.
Conceptual meaning
A formula is only one representation of a function; graphs, tables, mappings, and verbal rules may describe the same relationship. Calculus statements are always tied to domain: limits approach domain boundaries, derivatives require nearby inputs, and integrals accumulate across intervals.
A dependable method and decision rule
- Determine the natural domain before simplifying.
- Identify transformations from a familiar parent function.
- Find intercepts and use algebra or symmetry to describe shape.
- For composition, work from the inside outward and enforce both domain conditions.
- For an inverse, verify one-to-one behavior or restrict the domain first.
