{
  "title": "Function composition analyzer",
  "human_url": "https://happytails.ai/en/calculators/function-composition-analyzer/",
  "agent_url": "https://happytails.ai/agents/en/calculators/function-composition-analyzer/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/function-composition-analyzer.md",
  "content": "Function composition analyzer\nBuild 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.\nHow to use Function composition analyzer\nType the outer function f(x), such as sqrt(x).\nEnter the inner function g(x), such as x^2 - 4.\nRun the analysis and compare both compositions, their simplified forms and their domains.\nDownload the JSON report or the CSV of spot checks.\nHow can we improve this tool?\nReport a problem or share an idea.\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\nUseful next steps\nGraphing calculator\nScientific calculator\nQuadratic formula calculator\nEach 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.\nHow the composition is built\nThe 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).\nExact domains\nConditions 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.\nHidden exclusions\nSimplifying 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.",
  "content_format": "text/plain",
  "visibility": "public",
  "links": [
    {
      "name": "Graphing calculator",
      "url": "https://happytails.ai/en/calculators/graphing-calculator/"
    },
    {
      "name": "Scientific calculator",
      "url": "https://happytails.ai/en/calculators/scientific-calculator/"
    },
    {
      "name": "Quadratic formula calculator",
      "url": "https://happytails.ai/en/calculators/quadratic-formula/"
    }
  ],
  "tool": {
    "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"
    }
  },
  "markdown": "# Function composition analyzer\n\n- Human page: https://happytails.ai/en/calculators/function-composition-analyzer/\n- Markdown: https://happytails.ai/en/calculators/function-composition-analyzer.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/function-composition-analyzer/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `function-composition-analyzer` |\n| Category | calculators |\n| Implementation | Registered implementation. Registration alone is not a production-quality guarantee. |\n| Browser execution | Use the visible action or interactive controls. |\n| Browser processing | Browser; see tool notes for network behavior. |\n| HTTP API | Not callable through HTTP. |\n| Browser-agent interface | run_current_tool |\n\n### Implementation notes\n\nSubstitutes exactly on your device and solves the domain of each composition, for expressions up to 1,000 characters.\n\n### Inputs and settings\n\nText input: Outer function f(x).\n\nExample or placeholder shown in the interface (not a validated fixture):\n\n```text\nsqrt(x)\n```\n\nThe function applied last, such as sqrt(x) or 1/x. Up to 1,000 characters. Review the settings before running.\n\n| Setting | Meaning | Control type | Default | Constraints |\n| --- | --- | --- | --- | --- |\n| `g` | Inner function g(x) | text | \"x + 1\" |  |\n\n### Browser-agent workflow\n\n1. Open the human page in a browser that supports WebMCP.\n2. Discover the interface exposed by that page; use its actual schema.\n3. Select files on the page first when the tool requires files.\n4. Call `run_current_tool` with valid input and settings.\n5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links.\n\nCalls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution.\n\n### Planning specification\n\nEach 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.\n\nThis describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support.\n\n### Complete machine-readable capability record\n\n```json\n{\n  \"id\": \"function-composition-analyzer\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"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.\",\n  \"implementation\": {\n    \"fields\": [\n      {\n        \"key\": \"g\",\n        \"label\": \"Inner function g(x)\",\n        \"value\": \"x + 1\",\n        \"type\": \"text\",\n        \"maxLength\": 1000,\n        \"help\": \"The function applied first in f(g(x)), in the same variable as f, such as x + 1 or 1/(x - 2).\"\n      }\n    ],\n    \"presentation\": \"metrics\",\n    \"inputCodeDirection\": \"ltr\",\n    \"registrationFamily\": \"exact-math-functions\",\n    \"inputMaxLength\": 1000,\n    \"inputErrorCode\": \"EXACT_MATH_FUNCTIONS_EXPRESSION_TOO_LONG\",\n    \"note\": \"Substitutes exactly on your device and solves the domain of each composition, for expressions up to 1,000 characters.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": true,\n      \"inputMaxLength\": 1000,\n      \"inputMode\": \"text\",\n      \"label\": \"Outer function f(x)\",\n      \"placeholder\": \"sqrt(x)\",\n      \"help\": \"The function applied last, such as sqrt(x) or 1/x. Up to 1,000 characters. Review the settings before running.\",\n      \"helpParts\": [\n        \"The function applied last, such as sqrt(x) or 1/x.\",\n        \"Up to 1,000 characters.\",\n        \"Review the settings before running.\"\n      ],\n      \"action\": null\n    },\n    \"mode\": \"explicit\",\n    \"interactive\": false\n  },\n  \"execution\": \"Browser interface\",\n  \"api\": null,\n  \"browser_agent\": {\n    \"supported\": true,\n    \"name\": \"run_current_tool\",\n    \"discovery\": \"WebMCP on the human page in a supporting browser\",\n    \"verification\": \"See audit/API-AUDIT.md; availability is not verification\"\n  }\n}\n```\n\n## Page guide\n\nBuild 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.\n\n### How to use Function composition analyzer\n\n1. Type the outer function f(x), such as sqrt(x).\n2. Enter the inner function g(x), such as x^2 - 4.\n3. Run the analysis and compare both compositions, their simplified forms and their domains.\n4. Download the JSON report or the CSV of spot checks.\n\nHow can we improve this tool?\n\nReport a problem or share an idea.\n\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\n\n## Useful next steps\n\n- [Graphing calculator](https://happytails.ai/en/calculators/graphing-calculator/) ([Markdown](https://happytails.ai/en/calculators/graphing-calculator.md))\n\n- [Scientific calculator](https://happytails.ai/en/calculators/scientific-calculator/) ([Markdown](https://happytails.ai/en/calculators/scientific-calculator.md))\n\n- [Quadratic formula calculator](https://happytails.ai/en/calculators/quadratic-formula/) ([Markdown](https://happytails.ai/en/calculators/quadratic-formula.md))\n\nEach 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.\n\n## How the composition is built\n\nThe 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).\n\n### Exact domains\n\nConditions 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.\n\n### Hidden exclusions\n\nSimplifying 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.\n"
}