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  "body": "\n## Definition\n\nLet $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]].\nA left inverse of $f$ is a function $g \\colon B \\to A$ such that $g \\circ f = \\mathrm{id}_A$, that is, $g(f(a)) = a$ for each $a \\in A$.\n\n## Remarks\n\nA function with a nonempty domain has a left inverse exactly when it is [[injective|0PVG1CRAJN]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].\nA left inverse is in general not unique. Its values on $B \\setminus f(A)$ can be any elements of $A$.\nIf $g$ is a left inverse of $f$, then $f$ is a [[right inverse|0DDPZAG5ER]] of $g$.\n\n## Explanations\n\n### L1 human\n\nLet $f \\colon \\mathbb{N} \\to \\mathbb{N}$ be $f(n) = n + 1$.\nThe function $g$ with $g(n) = n - 1$ for $n \\ge 1$ and $g(0) = 0$ is a left inverse: $g(f(n)) = n$.\nBut $f(g(0)) = 1 \\neq 0$, so $g$ is not an inverse on both sides.\n",
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    "name": "Left inverse",
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        "locator": "Left inverses",
        "url": "https://en.wikipedia.org/wiki/Inverse_function"
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  "markdown": "https://math2.tamnd.com/node/02486SD328.md",
  "msc": [
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  "name": "Left inverse",
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