Source: https://guthlabs.ai/docs/instruments/function-domain-range-analyzer/mechanism/

instrument mechanism / verified-clean

# Function Domain/Range Analyzer mechanism

The inspectable method record for Function Domain/Range Analyzer, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## 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

[01**Overview**](<https://guthlabs.ai/docs/instruments/function-domain-range-analyzer/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/function-domain-range-analyzer/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/function-domain-range-analyzer/qualification/>)

## 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`