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  "body": "\n## Definition\n\nLet $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]] and let $T$ be a [[subset|0FE8Q5S42Y]] of $B$.\nThe preimage of $T$ under $f$ is the set\n$$f^{-1}(T) = \\{a \\in A \\mid f(a) \\in T\\}.$$\n\n## Remarks\n\nAnother name is the inverse image. The notation $f^{-1}(T)$ makes sense for every $f$, also when $f$ has no [[inverse function|0W766JAPSN]].\nPreimages respect all set operations: $f^{-1}(T_1 \\cap T_2) = f^{-1}(T_1) \\cap f^{-1}(T_2)$, and the same holds for unions and complements.\nExample: for $f(x) = x^2$ on $\\mathbb{R}$, $f^{-1}([1, 4]) = [-2, -1] \\cup [1, 2]$.\n\n## Explanations\n\n### L1 human\n\nThe preimage answers the question: which inputs land in $T$?\nFor the function that sends a student to their grade, the preimage of $\\{A\\}$ is the set of students with grade $A$.\n",
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