Function Domain/Range Analyzer mechanism

Model scope

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.

• 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.

Units and conventions

• 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.

Inspectable vector

One of 4 source-bound review vectors:

{
  "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."
}

Failure boundary

• 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.

Other records

Sources and evidence

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

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