Book Isle piecewise functions: select the matching condition before substitution

A piecewise definition does not ask you to apply every formula under its brace. Read an original three-part definition in Book Isle, select conditions and keep a reproducible substitution record. The task is how one input matches a definition, beyond checking interval endpoints on a number line. The app does not select formulas for you or prove that a function is valid.

Keep each condition paired with its formula

Record the file, edition, chapter and a phrase beside the definition. Our original example says: for an input below zero, output its negative; at zero, output 2; above zero, output the input plus 1. Label these branches A, B, C, retaining each formula with its condition rather than copying a formula list. If wrapping obscures which condition belongs to a line, inspect the original page instead of guessing from vertical proximity. Copy the function name and input letter too; a different function in the same chapter must not enter this record. Mark an unstated input domain as unstated.

Test the condition before substituting

For input minus 3, first ask whether it is below zero, equal to zero or above zero. Only condition A holds, so the output is the negative of minus 3, namely 3. Mark the other branches unmatched; minus 3 plus 1 giving minus 2 is not a second answer. The minus sign in branch A takes an opposite value; it does not declare that every output is negative. If nested minus signs are easy to miss, place parentheses around the substituted input and retain the intermediate step. Separate condition selection from calculation to locate whether an error came from the branch or the arithmetic.

Do not borrow a neighbouring branch at zero

At input zero, only condition B holds and the output is 2. Branch A requires strictly below zero and C strictly above it; their proximity to zero does not move their results 0 or 1 onto zero. Use positive 3 to check C, obtaining 4. Record inputs minus 3, 0, 3 with outputs 3, 2, 4, without confusing input order with sorted output magnitudes. These samples test only the selected inputs. They do not establish that every real number was checked; coverage must also be considered from the conditions as a whole, rather than guaranteed by trying several more values.

Separate a point value from nearby behaviour

For minus 0.1 and positive 0.1, the original definition gives 0.1 and 1.1, while the value at zero remains 2. Different neighbouring and point values do not automatically invalidate a definition; they show why a nearby trend must not be substituted for the point value. Three numbers do not prove a limit or continuity. If the textbook discusses those concepts, read its separate definitions and argument. Label nearby samples separately so they are not copied as the required value at zero, and do not let a connecting graph line hide an explicitly specified point.

Record a gap after removing the middle branch

Create a clearly labelled control omitting the middle branch and retaining only below-zero and above-zero conditions. Zero then has no matching condition. If the declared domain is all real numbers, the control omits zero; if the domain explicitly excludes zero, discuss that new domain rather than declaring every gap an error. Record the original domain, control domain and zero status. Do not supply 0 or 2 yourself: undefined is different from a calculated zero, and a blank is not a numerical answer. Never save the control back into the original book as if the textbook had omitted the branch.

Compare actual outputs where conditions overlap

In another control, change A to below or equal to zero while retaining B, which outputs 2 at zero. At zero, A gives 0 and B gives 2: one input has inconsistent outputs. The upper line does not win unless the text provides an explicit priority rule. Conversely, overlapping formulas that agree throughout their overlap should not be called numerically conflicting merely because conditions intersect. Record the number of matching branches separately from the number of distinct outputs, distinguishing redundant expression, inconsistent definition and an explicit priority convention. The original definition’s consistency cannot be transferred to the modified control.

Troubleshoot condition signs and lost wrapping

If less-than and less-than-or-equal signs are unclear, enlarge that location and compare an available reliable original; do not infer a sign from the answer. After copying notes, check branch order, equality and minus signs. Plain-text line breaks may not preserve formula layout. If reliable text search is unavailable, locate the definition manually using its chapter and neighbouring phrase, without claiming Book Isle parsed its mathematics. When a printed answer disagrees, first verify the input, domain and whether you opened a revised edition, then state a specific question instead of immediately rewriting the author’s formula.

Exit and return to review the selection record

Exit the reader and return to the original definition. Cover your results and select branches again for minus 3, 0, 3 before checking the recorded 3, 2, 4. Save six fields: input, satisfied condition, chosen formula, substitution steps, output and edition. Place the gap and conflict controls after the original example with clear labels. Explain why zero cannot borrow a neighbouring branch and what changes when a condition sign is modified. The result is a reviewable reading of branch selection, not a sample-based proof that every complex piecewise function is continuous or defined everywhere.

Related guides and sources

Book Isle: Align interval brackets, inequalities and number-line endpoints; Read function composition with Book Isle: pass the inner output to the outer function.

Checked on November 10, 2026: Apple public version information lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the official App Store for feature descriptions; specific controls must still be checked in the installed version.

Method reference: OpenStax: Domain and Range. Numbers, short passages and controls are original exercises. Sources explain concepts rather than prove automatic recognition or grading in the app. Manual notes may be kept externally. This article does not claim step-by-step testing on a physical device or private server; unresolved content should remain marked for review.