Read even and odd functions in Book Isle: check the domain before negative inputs

A cubic exponent does not permit ignoring the domain and labelling a function odd. Read original function cards in Book Isle, separating input range, substitution of negative inputs and negation of the whole output. Check intuition with counterexamples. Keep paper reasoning separate from reader navigation, with optional external notes; neither automatic classification nor visual resemblance is a proof.

Attach a domain to every function card

Make three original cards: f(x)=x^3-x, g(x)=x^2+1 and h(x)=x+1, initially on all real numbers. Record book or file, location and function letter, not just the formula. Check whether every allowed x also permits minus x before comparing outputs. No graphing feature is required: use paper cards and return to their reader pages. If the book restricts the domain, preserve that restriction rather than silently extend it to all real numbers.

Treat both equalities as domain-wide conditions

An even function has the same output at x and minus x for every domain input; an odd function has opposite outputs for every such input. Separate f(-x) from -f(x): one changes the input first, the other negates the entire original output. Testing zero or one positive value does not establish a universal statement, and one equal pair is not all pairs. Keep definitions beside examples and label rules separately from trial calculations for later checking.

Expand the cubic at a negative input

Replace every x in A with parenthesised minus x. Its cube is negative x cubed and subtracting minus x gives plus x, matching the negation of the whole original formula. Thus A is odd on the real numbers. At two it gives six; at minus two, minus six. Do not apply substitution only to the first term or miscopy subtraction of minus x. This numerical pair checks transcription; the symbolic identity establishes all inputs, so retain their distinct roles.

Keep the constant when checking the square

For B, squaring minus x returns x squared and the constant plus one stays unchanged. Equal outputs establish evenness. Both two and minus two give five, while zero gives one: an even function need not pass through the origin. Changing plus one to minus one confuses input substitution with negation of the whole function. Return to the two-operation record. Domain and algebra establish the claim, without pixel symmetry or an assumed in-app curve-reflection tool.

Use counterexamples to reject both labels

For C, one gives two and minus one gives zero. These are neither equal nor opposites, so C is neither even nor odd. One permitted counterexample disproves a universal condition; testing all real numbers is unnecessary. A nonzero value at zero also rules out oddness when zero belongs to the domain, but zero output at zero alone does not prove oddness. Record precisely which label the counterexample rejects without claiming the function has no other symmetry or property.

Change only the domain and reassess

Save D with A’s identical formula but a strictly positive domain. Two is permitted and minus two is not. Under the symmetric-domain convention used here, D cannot inherit A’s odd label; evaluating an excluded minus two is not a valid check. The cubic term has not changed, but the function includes its domain. Do not add zero or confuse positive with nonnegative inputs to obtain a desired answer. Name the control explicitly so reopening does not return to A unnoticed.

Check the shortcut based on exponents

If a cubic exponent alone suggests oddness, test C or add a constant one to A: the constant breaks the required identity. A square alone likewise does not establish evenness if a linear term is present. Diagnose omitted terms, ignored domains or sign expansion instead of renaming functions. The zero function on a symmetric domain satisfies both identities, so the labels are not universally exclusive. Return and enlarge unclear text; retain uncertainty instead of attributing a guess to the book or app.

Reopen and reconstruct all four cards

Leave the reader and reopen from the shelf, checking file, location and domain wording before reconstructing A through D without the conclusions. Retain symbolic comparisons, the two/minus-two examples, C’s counterexample and D’s excluded inputs. Each label must trace to the rule and actual domain. Existing bookmarks may help return, but no cross-device automatic verification is assumed. Deliver reproducible reasoning and reassess after edition, domain or formula changes rather than permanently classify every function with the same name.

Related guides and sources

Book Isle piecewise functions: select the matching condition before substitution; Read function composition with Book Isle: pass the inner output to the outer function.

Checked on November 14, 2026: Apple public information lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the official App Store for published capabilities; verify specific controls in the installed version.

Method reference: OpenStax: transformations and even or odd functions. Samples, passages and controls are original exercises. Sources establish definitions and checking methods rather than new automatic analysis in the app. Manual records may be kept externally, and unknown fields should remain explicit. This article does not claim step-by-step testing on a physical device, private server or every output environment.