happytails.ai

Function composition analyzer

Open Function composition analyzer on Happytails.ai. This view has the same page content and links, with explicit tool capabilities. Read the processing notes before using a tool.

Tool capabilities

{
  "id": "function-composition-analyzer",
  "category": "calculators",
  "built": true,
  "requirements": "Each function is limited to 1,000 characters, one variable, 24 nesting levels and 100 digits per number; exact domains support polynomial and rational conditions with up to 4 abs terms and factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of spot checks.",
  "implementation": {
    "fields": [
      {
        "key": "g",
        "label": "Inner function g(x)",
        "value": "x + 1",
        "type": "text",
        "maxLength": 1000,
        "help": "The function applied first in f(g(x)), in the same variable as f, such as x + 1 or 1/(x - 2)."
      }
    ],
    "presentation": "metrics",
    "inputCodeDirection": "ltr",
    "registrationFamily": "exact-math-functions",
    "inputMaxLength": 1000,
    "inputErrorCode": "EXACT_MATH_FUNCTIONS_EXPRESSION_TOO_LONG",
    "note": "Substitutes exactly on your device and solves the domain of each composition, for expressions up to 1,000 characters.",
    "experience": {
      "automatic": false,
      "single": true,
      "inputMaxLength": 1000,
      "inputMode": "text",
      "label": "Outer function f(x)",
      "placeholder": "sqrt(x)",
      "help": "The function applied last, such as sqrt(x) or 1/x. Up to 1,000 characters. Review the settings before running.",
      "helpParts": [
        "The function applied last, such as sqrt(x) or 1/x.",
        "Up to 1,000 characters.",
        "Review the settings before running."
      ],
      "action": null
    },
    "mode": "explicit",
    "interactive": false
  },
  "execution": "Browser interface",
  "api": null,
  "browser_agent": {
    "supported": true,
    "name": "run_current_tool",
    "discovery": "WebMCP on the human page in a supporting browser",
    "verification": "See audit/API-AUDIT.md; availability is not verification"
  }
}

Page guide

Build f(g(x)) and g(f(x)) exactly and find the domain of each composition. The report shows values that the simplified formula would wrongly accept.

How to use Function composition analyzer
  1. Type the outer function f(x), such as sqrt(x).
  2. Enter the inner function g(x), such as x^2 - 4.
  3. Run the analysis and compare both compositions, their simplified forms and their domains.
  4. Download the JSON report or the CSV of spot checks.

Useful next steps

Graphing calculatorScientific calculatorQuadratic formula calculator

Each function is limited to 1,000 characters, one variable, 24 nesting levels and 100 digits per number; exact domains support polynomial and rational conditions with up to 4 abs terms and factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of spot checks.

How the composition is built

The tool replaces every x in f with the whole expression g, and the other way round. The domain of f(g(x)) is every x in the domain of g whose value g(x) lies in the domain of f, so the conditions of both functions are kept. Method: OpenStax, College Algebra 2e, section 3.4 Composition of Functions (2021).

Exact domains

Conditions such as denominators not equal to zero, square roots of nonnegative values and logarithms of positive values are solved exactly with sign charts, including irrational endpoints. When a condition is not a polynomial or rational condition, the conditions are listed without an interval.

Hidden exclusions

Simplifying 1/(1/(x - 1)) to x - 1 hides the excluded value x = 1. The report lists such points so the simplified formula is not used where the composition is undefined. The spot checks are decimal evidence, not proof.