Even and odd functions

Even and odd functions: algebra and graph symmetry

Classify functions by checking the relationship between opposite inputs. This precalculus practice set connects algebraic substitution with reflection across the vertical axis and symmetry about the origin. Polynomial, rational and piecewise rules, finite observations and an original coordinate graph provide varied evidence to assess.

A symmetric domain is essential: a familiar even formula can lose the usual classification when its domain is restricted asymmetrically. Other questions address the zero function, the value at zero for an odd function when that input is allowed, and an odd function whose domain excludes zero. You will also distinguish a few matching sample values from an identity that holds everywhere.

Prerequisites

Review function notation, negative-input substitution, polynomial arithmetic and piecewise rules. First check the domain, then simplify the output at the opposite input. Compare the result with the original output and its negative; do not rely only on how a formula looks.

Practice and feedback

Single-select and select-all tasks progress toward case reasoning and parameter conditions. The explanations include a direct identity, a paired point or a counterexample. State the domain when justifying your answer, and treat a plotted pattern as evidence to examine rather than an excuse to skip the algebraic definition.