{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nMany functions given by algebraic formulas possess a formula for their inverse. This is because the inverse $f^{-1}$ of an invertible function $f\\colon\\mathbb{R}\\to\\mathbb{R}$ has an explicit description as\n\n$$f^{-1}(y)=(\\text{the unique element }x\\in \\mathbb{R}\\text{ such that }f(x)=y).$$\n\nThis allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if $f$ is the function\n\n$$f(x) = (2x + 8)^3$$\n\nthen to determine $f^{-1}(y)$ for a real number $y$, one must find the unique real number $x$ such that $(2x + 8)^{3} = y$. This equation can be solved:\n\n$$\\begin{aligned} y \u0026 = (2x+8)^3 \\\\ \\sqrt[3]{y} \u0026 = 2x + 8 \\\\ \\sqrt[3]{y} - 8 \u0026 = 2x \\\\ \\dfrac{\\sqrt[3]{y} - 8}{2} \u0026 = x . \\end{aligned}$$\n\nThus the inverse function $f^{\\,-1}$ is given by the formula\n\n$$f^{-1}(y) = \\frac{\\sqrt[3]{y} - 8} 2.$$\n\nSometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if $f$ is the function\n\n$$f(x) = x - \\sin x ,$$\n\nthen $f$ is a bijection, and therefore possesses an inverse function $f^{\\,-1}$. The formula for this inverse has an expression as an infinite sum:\n\n$$f^{-1}(y) = \\sum_{n=1}^\\infty \\frac{y^{n/3}}{n!} \\lim_{ \\theta \\to 0} \\left( \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d} \\theta^{\\,n-1}} \\left( \\frac \\theta { \\sqrt[3]{ \\theta - \\sin( \\theta )} } \\right)^n \\right).$$\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "inventory": [
      "ZADBHJ84M0"
    ],
    "kind": "statement",
    "msc": [
      "03E20"
    ],
    "name": "Formula for the inverse",
    "role": "proposition",
    "sources": [
      {
        "locator": "Inverse function, section Examples \u003e Formula for the inverse",
        "mention": "wikipedia://en/14907#8I5T",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:14907@r1368588964"
      }
    ],
    "tag": "QZWN74YRMM"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/QZWN74YRMM/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/QZWN74YRMM.md",
  "msc": [
    "03E20"
  ],
  "name": "Formula for the inverse",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QZW/QZWN74YRMM.md",
  "status": null,
  "tag": "QZWN74YRMM",
  "uri": "m2://QZWN74YRMM",
  "url": "https://math2.tamnd.com/wiki/formula-for-the-inverse/",
  "writing": {
    "tag": "QZWN74YRMM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}