Function Domain/Range Analyzer

Inspectable records

Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Area

Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

Tranche

tranche-8

Evidence ID

t8-035

Ledger rows

1

No ledger note is attached.

Review findings

The stated restriction rules are correct under real-valued, radian-based assumptions, but the range method is underspecified and mathematically unreliable for the claimed generality. Exact symbolic range analysis, or a clearly bounded estimate with deterministic sampling parameters and explicit uncertainty labeling, is required before this instrument can be marked PASS.

Formula and scope findings

• The denominator≠0, radicand≥0, and log-argument>0 rules are valid real-domain rules, but they are not sufficient for all claimed common function types. Logarithm bases also require base>0 and base≠1.
• The radicand≥0 rule applies to even roots; applying it to odd roots would be incorrect. Nested expressions require restrictions to be propagated through every subexpression.
• Numeric range sampling is not a standard reliable range algorithm: it can miss holes, isolated values, extrema, unbounded behavior, and intervals near poles. The exact range cannot be determined from the stated sampling method.
• The specification does not define a sampling interval, density, adaptive strategy, pole detection method, tolerance, or representation for unbounded ranges. Therefore the estimated range is not deterministic or reproducible.
• The output claims domain and range determination while the honest label states that range is only estimated numerically for complex functions; the output must clearly distinguish formally derived ranges from estimates.

Unit assumptions

• Inputs are real-valued functions of a real variable x; complex domains and ranges are excluded unless explicitly supported.
• Trigonometric arguments are assumed to be in radians. If degree input is supported, conversion must be explicit.
• Logarithm arguments and bases are dimensionless; logarithm bases, when present, must satisfy b>0 and b≠1.
• Interval notation uses closed brackets for included finite endpoints, parentheses for excluded endpoints or infinity, and unions for disconnected sets.

Boundary hardening

Recorded hazards

• Division by zero can occur if a sampler evaluates a rational function at an excluded denominator root.
• Logarithm evaluation must reject argument≤0 and must validate the base when a non-natural base is supplied.
• Even-root evaluation must reject negative radicands; odd roots do not have the same restriction.
• Floating-point tolerance near zeros, poles, and interval endpoints can incorrectly include or exclude domain points.
• Sampling finite windows cannot establish behavior as x approaches infinity or a pole.
• Periodic functions require symbolic periodicity or a justified global analysis; finite sampling can miss extrema and discontinuities.
• Removable discontinuities can be lost if expressions are algebraically simplified before domain restrictions are retained.
• The implementation needs a deterministic sampling specification and must label sampled ranges as estimates rather than exact results.

Errata and corrections

No applied erratum is attached to this unique-name record.

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "inputs": {
    "function": "(2*x + 1)/(x - 3)"
  },
  "expected_outputs": {
    "canonical_domain": "(-∞, 3) ∪ (3, ∞)",
    "canonical_range": "(-∞, 2) ∪ (2, ∞)",
    "estimated_range": "Sampling-dependent; the exact range is (-∞, 2) ∪ (2, ∞).",
    "restrictions": [
      "x - 3 ≠ 0",
      "x ≠ 3"
    ],
    "spot_checks": [
      {
        "at": 0,
        "input_side": -0.3333333333,
        "answer_side": -0.3333333333
      },
      {
        "at": 4,
        "input_side": 9,
        "answer_side": 9
      }
    ]
  },
  "reference_data": {
    "range_derivation": "Let y=(2x+1)/(x-3). Then y(x-3)=2x+1, so x(y-2)=3y+1. For y=2 this gives 0=7, impossible; every other real y has x=(3y+1)/(y-2)."
  },
  "working": "Domain: set the denominator nonzero: x-3≠0, so x≠3. Domain is (-∞,3)∪(3,∞). Range: solve y=(2x+1)/(x-3). Multiply: yx-3y=2x+1. Rearrange: x(y-2)=3y+1. If y=2, the equation becomes 0=7, impossible. If y≠2, x=(3y+1)/(y-2) is real, so every y except 2 occurs. Therefore range is (-∞,2)∪(2,∞). Spot check at x=0: (2·0+1)/(0-3)=1/(-3)=-1/3=-0.3333333333. Spot check at x=4: (2·4+1)/(4-3)=9/1=9."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json · VERIFIED_CLEAN
• evidence-tranche8/tranche8-results.json